2010 Mathematics Subject Classification. 54J05, 54A05, 54D99,
54E05, 54E17, 54E99
Key words and phrases. algebraic general topology, limit, funcoid,
differential equations
Abstract. I consider (generalized) limit of arbitrary (discontin-
uous) function, defined in terms of funcoids. Definition of general-
ized limit makes it obvious to define such things as derivative of an
arbitrary function, integral of an arbitrary function, etc. It is given
a definition of non-differentiable solution of a (partial) differential
equation. It’s raised the question how do such solutions “look like”
starting a possible big future research program.
The generalized solution of one simple example differential
equation is also considered.
The generalized derivatives and integrals are linear operators.
For example
R
b
a
f(x)dx −
R
b
a
f(x)dx = 0 is defined and true for
every function.
Contents
1. Introduction 4
2. A popular explanation of generalized limit 4
3. Funcoids 7
4. Limit for funcoids 8
5. Axiomatic generalized limit 10
6. Generalized limit 11
7. Generalized limit vs axiomatic generalized limit 16
8. Operations on generalized limits 17
9. Equivalence of different generalized limits 22
10. Hierarchy of singularities 23
11. Funcoid of singularities 24
12. Funcoid of supersingularities 25
13. Example differential equation 26
Bibliography 29
3
4 CONTENTS
1. Introduction
I defined funcoid and based on this generalized limit of an arbitrary
(even discontinuous) function in [1].
In this article I consider generalized limits in more details.
This article is written in such a way that a reader could understand
the main ideas on generalized limits without resorting to reading [1]
beforehand, but to follow the proofs you need read that first.
Definition of generalized limit makes it obvious to define such things
as derivative of an arbitrary function, integral of an arbitrary function,
etc.
Note that generalized limit is a “composite” object, not just a sim-
ple real number, point, or “regular” vector.
2. A popular explanation of generalized limit
For an example, consider some real function f from x-axis to y-axis:
−2 −1 1 2
−5
5
Take it’s infinitely small fragment (in our example, an infinitely
small interval for x around zero; see below for an explanation what is
infinitely small):
2. A POPULAR EXPLANATION OF GENERALIZED LIMIT 5
0.5 1 1.5 2
2
4
6
Next consider that with a value y replaced with an infinitely small
interval like [y − ; y + ]:
0.5 1 1.5 2
2
4
6
Now we have “an infinitely thin and short strip”. In fact, it is
the same as an “infinitely small rectangle” (Why? So infinitely small
behave, it can be counter-intuitive, but if we consider the above medi-
tations formally, we could get this result):
6 CONTENTS
0.5 1 1.5 2
2
4
6
This infinitely small rectangle’s y position uniquely characterizes
the limit of our function (in our example at x → 0).
If we consider the set of all rectangles we obtain by shifting this
rectangle by adding an arbitrary number to x, we get
0.5 1 1.5 2
2
4
6
Such sets one-to-one corresponds to the value of the limit of our
function (at x → 0): Knowing such the set, we can calculate the limit
(take its arbitrary element and get its so to say y-limit point) and
knowing the limit value (y), we could write down the definition of this
set.
So we have a formula for generalized limit:
lim
x→a
f(x) = {X ◦ f|
∆(a)
◦ r | r ∈ G}
3. FUNCOIDS 7
where G is the group of all horizontal shifts of our space R, f|
∆(a)
is the
function f of which we are taking limit restricted to the infinitely small
interval ∆(a) around the point a, X◦ is “stretching” our function graph
into the infinitely thin “strip” by applying a topological operation to
it.
What all this (especially “infinitely small”) means? It is filters and
“funcoids” (see below for the definition).
Why we consider all shifts of our infinitely small rectangle? To make
the limit not dependent of the point a to which x tends. Otherwise the
limit would depend on the point a.
Note that for discontinuous functions elements of our set (our limit
is a set) won’t be infinitely small “rectangles” (as on the pictures), but
would “touch” more than just one y value.
The interesting thing here is that we can apply the above formula
to every function: for example to a discontinuous function, Dirichlet
function, unbounded function, unbounded and discontinuous at every
point function, etc. In short, the generalized limit is defined for every
function. We have a definition of limit for every function, not only a
continuous function!
And it works not only for real numbers. It would work for example
for any function between two topological vector spaces (a vector space
with a topology).
Hurrah! Now we can define derivative and integral of every func-
tion.
3. Funcoids
I will reprise (without proofs) several equivalent definitions of fun-
coid from [1]:
Binary relation δ between two sets (source and destination of the
funcoid), conforming to the axioms:
(1) not ∅ δ X
(2) not X δ ∅
(3) I ∪ J δ K ⇐⇒ I δ K ∧ J δ K
(4) K δ I ∪ J ⇐⇒ K δ I ∧ K δ J
Pair of functions (α, β) between the sets of filters filters on some two
sets (source and destination of the funcoid), conforming to the formula:
α(X ) u Y 6= ⊥ ⇐⇒ β(Y) u X 6= ⊥.
Remark 1. Funcoid (α, β) is determined by the value of α (or value
of β).
8 CONTENTS
A function ∆ from the set of subsets of some set (source of the
funcoid) to the set of filters on some set (destination of the funcoid),
conforming to the axioms:
(1) ∆(∅) = ⊥
(2) ∆(X t Y ) = ∆(X) t ∆(Y )
(Here t and u are the join and the meet correspondingly on the
lattice of filters with order reverse to set-theoretic inclusion, ⊥ is the
improper filter.)
Note that we define things to have the equations:
(1)
X [f]
∗
Y ⇔ X δ Y ⇔
↑ X [(α, β)] ↑ Y ⇔ α(X) u Y 6= ⊥ ⇔
β(Y ) u X 6= ⊥
(2) hfi
∗
X = ∆X = α ↑ X
We will denote partial orders as v.
I will call endofuncoid a funcoid whose source and destination are
the same.
Funcoids form a semigroup (or precategory, dependently on the
exact axioms) with the operation defined by the formula:
(α
1
, β
1
) ◦ (α
0
, β
0
) = (α
1
◦ α
0
, β
0
◦ β
1
).
We denote h(α, β)i = α and (α, β)
−1
= (β, α).
Funcoids also form a poset which is a complete lattice.
Funcoids are a generalization of both topological spaces and prox-
imity spaces (see [1]).
Also funcoids are ([1]) a generalization of binary relations. (I will
denote the funcoid corresponding ([1]) to a binary relation f as ↑ f)
This makes funcoids a common generalization for topologies/proximities
and functions, so they are a convenient tool to study functions between
spaces.
Another important for this article operation on funcoids is restrict-
ing a funcoid to a filter (generalizing restricting a function to a set): f|
X
for a funcoid f and filter X .
In [1] we also have a funcoid called funcoidal product X ×
FCD
Y of
two filters X and Y.
4. Limit for funcoids
The following is a straigthforward generalization of the well known
concepts of adherent point of a set (more generally a cluster point of a
4. LIMIT FOR FUNCOIDS 9
filter), a limit point of a filter, a and limit of a function in a topological
space.
Note 2. Due to an unfortunate choice of terminology, limit point
of a filter is not a generalization of a limit point of a set. Limit point
for a set isn’t a beautiful term and we won’t use it (in this work), so
by limit point we will always mean a limit point of a filter.
So, generalizing the corresponding concepts for topological spaces:
Let d be a funcoid.
Definition 3. The set of adherent points of A is Cor hd
−1
i A or
what is the same
n
x
hdi{x}6A
o
.
Proposition 4. There exists a (unique) funcoid A such that hAi A
is exactly the set of adherent points of A.
Proof. Prove it directly or using that Cor is a component of a
funcoid.
Definition 5. Limit point of a filter A on Dst X is an x ∈ Dst A
such that hdi
∗
{x} w A.
Proposition 6. If d is reflexive, then there exists a (unique) “dual
funcoid” (a pointfree funcoid) L : Ob d → (Ob d)
dual
such that hLi A is
exactly the set of limit points of A.
Proof. The set of limit points of the empty set is the maximal set.
The set of limit points of A ∪ B (for sets A, B) is the set of points x
such that hdi {x} w A ∪ B that is hdi {x} w A and hdi {x} w B that is
the intersection of the set of limit points of A and B.
Thus the set of limit points is a component of such a pointfree
funcoid.
Proposition 7. L and A coincide on ultrafilters.
Proof. Because hdi
∗
{x} 6 a ⇔ hdi
∗
{x} w a for an ultrafilter a.
We have shown that concepts of both limit points and adherent
points are essentially funcoids. In traditional general topology limit of
a function is defined using limit points of a filter. We will generalize it to
limit regarding an arbitrary funcoid (in place of the funcoid describing
limit points). We will call this arbitrary funcoid the point funcoid and
denote it X.
Definition 8. Limit of a funcoid f is the filter
lim f = im(X ◦ f).
10 CONTENTS
Definition 9. Limit of a funcoid f at a filter X is the filter
lim
X
f = lim f|
X
= hX ◦ fi X .
Remark 10. If X = L, then the limit is either an one-element of
an empty set (“no limit” in traditional topology).
In [1] limit for a funcoid f was defined this way: f tends to filter A
(f → A) regarding a funcoid X on a filter X iff
im f v hXi A.
lim f is such a point that f tends to lim f.
Proposition 11. That definition from [1] coincides with our above
definition, if X = L.
Proof. In this proof lim f will mean our definition, not the defi-
nition from [1]. We need to prove that lim f =
n
x
im f vhXi
∗
{x}
o
.
Really,
lim f = im(L ◦ f) = hLi im f =
x
hXi
∗
{x} w im f
.
If X is Hausdorff (T
2
-separable) (see [1]), then there exists no more
than one lim f.
5. Axiomatic generalized limit
Let X be a (fixed) funcoid. For example, X = d where d is some
proximity or d = A or d = L (up to a duality).
By definion lim
X
f = hX ◦ fi X (for every funcoid f).
Remark 12. If hXi y is an limit point (considered as an one-element
set) of y and f is a function, then the above defined lim is the same
as limit in traditional calculus and topology (except that it is an one-
element set of points instead of a point). Empty set means “no limit”.
Let some group G (e.g. the group of all shifts on a vector space, to
give an example) is fixed.
Definition 13. Axiomatic generalized limit is a two-arguments
function (f, X ) 7→ xlim
X
f from the set FCD(A, B) × F (A) to the set
of functions from from filters C such that for r ∈ G such that C v hri X
we have:
(xlim
X
f)C = lim
hr
−1
iC
f.
6. GENERALIZED LIMIT 11
Remark 14. Its meaning is that lim
C
f can be restored from xlim
X
f.
Proposition 15. To describe an axiomatic generalized limit, it’s
enough to define it on ultrafilters.
Proof. Easily follows from the fact that a funcoid is described by
its values on ultrafilters.
Thus axiomatic generalized limit gives a detailed behavior of a func-
tion at a filter (its limit at every its atomic subfilter).
Obvious 16. xlim
X
f = xlim
hriX
(f ◦ r
−1
) for every r ∈ G.
Theorem 17. lim
X
f can be restored knowing xlim
X
f.
Proof.
lim
X
f = hX ◦ fi X =
G
x∈atoms X
hX ◦ fi x =
G
x∈atoms X
lim x = (xlim
X
f)x.
Let y be an arbitrary point of the space Dst X. Consider the con-
stant function f whose value is this y. Then the first axiom above
determines the xlim
C
f for every filter C.
I will denote xlim
C
f = τ(y).
Remark 18. The above easily generalizes for y being a set of points.
Obvious 19. τ is an injection, if Src X is a non-empty set and hXi is
an injection on one-element sets.
Corollary 20. τ is an injection, if Src X is a non-empty set and
X = A and X is Hausdorff.
Definition 21. I will call axiomatic singularities all possible values
of axiomatic generalized limits.
In other words, on Hausdorff topologies the set of singularities with
non-empty domains is an extension of the set of points Dst X (up to a
bijection).
6. Generalized limit
6.1. The definition of generalized limit. In [1] generalized
limit is defined like the formula:
12 CONTENTS
(1) xlim f =
X ◦ f◦ ↑ r
r ∈ G
.
We suppose:
Let µ and X be endofuncoids (on sets Ob µ, Ob X). Let G be a
transitive permutation group on Ob µ.
We require that µ and every r ∈ G commute, that is
(2) µ◦ ↑ r =↑ r ◦ µ.
We require for every y ∈ Ob X
(3) X w hXi ↑ {y} ×
FCD
hXi ↑ {y}.
Proposition 22. Formula (3) follows from X w X ◦ X
−1
.
Proof. Let X w X ◦ X
−1
. Then
hXi ↑ {y} ×
FCD
hXi ↑ {y} =
X ◦ (↑ {y}×
FCD
↑ {y}) ◦ X
−1
=
X◦ ↑ ({y} × {y}) ◦ X
−1
v
X ◦ 1 ◦ X
−1
=
X ◦ X
−1
v X.
(Here 1 is the identity element of the semigroup of endofuncoids.)
So we have (generalized) limits of arbitrary functions acting from
Ob µ to Ob X. (The functions in consideration are not required to be
continuous.)
Remark 23. Most typically G is the group of translations of some
topological vector space
1
. So in particular we have defined limit of an
arbitrary function acting from a vector topological space to a topolog-
ical space.
6.2. Injection from the set of points to the set of all gen-
eralized limits. The function τ will define an injection from the set
of points of the space X (“numbers”, “points”, or “vectors”) to the set
of all (generalized) limits (i.e. values which xlim
x
f may take).
Definition 24.
τ(y)
def
=
hµi ↑ {x} ×
FCD
hXi ↑ {y}
x ∈ D
.
1
I remind that every Banach space, every normed space, and every Hilbert
space is a vector topological space.
6. GENERALIZED LIMIT 13
Proposition 25.
τ(y) =
(hµi ↑ {x} ×
FCD
hXi ↑ {y})◦ ↑ r
r ∈ G
for every (fixed) x ∈ D.
Proof.
(hµi ↑ {x} ×
FCD
hXi ↑ {y})◦ ↑ r =
↑ r
−1
hµi ↑ {x} ×
FCD
hXi ↑ {y} =
hµi
↑ r
−1
↑ {x} ×
FCD
hXi ↑ {y} =
hµi ↑ {r
−1
x} ×
FCD
hXi ↑ {y} ∈
hµi ↑ {x} ×
FCD
hXi ↑ {y}
x ∈ D
.
Reversely
hµi ↑ {x} ×
FCD
hXi ↑ {y} =
(hµi ↑ {x} ×
FCD
hXi ↑ {y})◦ ↑ e
where e is the identify element of G.
Proposition 26.
τ(y) = xlim(hµi ↑ {x}×
FCD
↑ {y})
(for every x). Informally: Every τ(y) is a generalized limit of a constant
function.
Proof.
xlim(hµi ↑ {x}×
FCD
↑ {y}) =
X ◦ (hµi ↑ {x}×
FCD
↑ {y})◦ ↑ r
r ∈ G
=
(hµi ↑ {x} ×
FCD
hXi ↑ {y})◦ ↑ r
r ∈ G
= τ(y).
Corollary 27. The τ defined in this section for generalized limits
“coincides” with the τ defined in the section about axiomatic general-
ized limits.
In further we will use on of the definitions of continuity from [1]:
f ∈ C(µ, X) ⇔ f ◦ µ v X ◦ f
and other notation from the book.
14 CONTENTS
Theorem 28. If f is a function and f|
hµi↑{x}
∈ C(µ, X) and hµi ↑
{x} w↑ {x} then xlim
x
f = τ(fx).
Proof. f|
hµi↑{x}
◦ µ v X ◦ f|
hµi↑{x}
v X ◦ f ; thus hfi hµi ↑ {x} v
hXi hfi ↑ {x}; consequently we have
X w hXi hfi ↑ {x} ×
FCD
hXi hfi ↑ {x} w
hfi hµi ↑ {x} ×
FCD
hXi hfi ↑ {x}.
X ◦ f|
hµi↑{x}
w
(hfi hµi ↑ {x} ×
FCD
hXi hfi ↑ {x}) ◦ f|
hµi↑{x}
=
(f|
hµi↑{x}
)
−1
hfi hµi ↑ {x} ×
FCD
hXi hfi ↑ {x} w
D
id
FCD
dom f |
hµi↑{x}
E
hµi ↑ {x} ×
FCD
hXi hfi ↑ {x} w
dom f|
hµi↑{x}
×
FCD
hXi hfi ↑ {x} =
hµi ↑ {x} ×
FCD
hXi hfi ↑ {x}.
im(X ◦ f|
hµi↑{x}
) = hXi hfi ↑ {x};
X ◦ f|
hµi↑{x}
v
hµi ↑ {x} ×
FCD
im(X ◦ f|
hµi↑{x}
) =
hµi ↑ {x} ×
FCD
hXi hfi ↑ {x}.
So X ◦ f|
hµi↑{x}
= hµi ↑ {x} ×
FCD
hXi hfi ↑ {x}.
Thus
xlim
x
f =
(hµi ↑ {x} ×
FCD
hXi hfi ↑ {x})◦ ↑ r
r ∈ G
=
τ(fx).
Remark 29. Without the requirement of hµi ↑ {x} w↑ {x} the
last theorem would not work in the case of removable singularity.
Theorem 30. Let X v X ◦ X. If f|
hµi↑{x}
X
→↑ {y} then xlim
x
f =
τ(y).
6. GENERALIZED LIMIT 15
Proof. im f|
hµi↑{x}
v hXi ↑ {y}; hfi hµi ↑ {x} v hXi ↑ {y};
X ◦ f|
hµi↑{x}
w
(hXi ↑ {y} ×
FCD
hXi ↑ {y}) ◦ f|
hµi↑{x}
=
(f|
hµi↑{x}
)
−1
hXi ↑ {y} ×
FCD
hXi ↑ {y} =
id
FCD
hµi↑{x}
◦f
−1
hXi ↑ {y} ×
FCD
hXi ↑ {y} w
id
FCD
hµi↑{x}
◦f
−1
hfi hµi ↑ {x} ×
FCD
hXi ↑ {y} =
id
FCD
hµi↑{x}
f
−1
◦ f
hµi ↑ {x} ×
FCD
hXi ↑ {y} w
id
FCD
hµi↑{x}
id
FCD
hµi↑{x}
hµi ↑ {x} ×
FCD
hXi ↑ {y} =
hµi ↑ {x} ×
FCD
hXi ↑ {y}.
On the other hand,
f|
hµi↑{x}
v hµi ↑ {x} ×
FCD
hXi ↑ {y};
X ◦ f|
hµi↑{x}
v hµi ↑ {x} ×
FCD
hXi hXi ↑ {y} v
hµi ↑ {x} ×
FCD
hXi ↑ {y}.
So X ◦ f|
hµi↑{x}
= hµi ↑ {x} ×
FCD
hXi ↑ {y}.
xlim
x
f =
X ◦ f|
hµi↑{x}
◦ ↑ r
r ∈ G
=
(hµi ↑ {x} ×
FCD
hXi ↑ {y})◦ ↑ r
r ∈ G
= τ(y).
Corollary 31. If lim
X
hµi↑{x}
f = y then xlim
x
f = τ(y) (provided
that X v X ◦ X).
We have injective τ if hXi ↑ {y
1
} u hXi ↑ {y
2
} = ⊥
F (Ob µ)
for every
distinct y
1
, y
2
∈ Ob X that is if X is T
2
-separable.
6.3. Hausdorff and Kolmogorov funcoids.
Definition 32. A funcoid f is Kolmogorov when hfi ↑ {x} 6=
hfi ↑ {y} for every distinct points x, y ∈ dom f.
Definition 33. Limit lim F = x of a filter F regarding funcoid f
is such a point that hfi ↑ {x} w F.
Definition 34. Hausdorff funcoid is such a funcoid that every
proper filter on its image has at most one limit.
16 CONTENTS
Proposition 35. The following are pairwise equivalent for every
funcoid f:
(1) f is Hausdorff.
(2) x 6= y ⇒ hf i ↑ {x} u hfi ↑ {y} = ⊥.
Proof.
1⇒2: If 2 does not hold, then there exist distinct points x and y such
that hfi ↑ {x} u hfi ↑ {y} 6= ⊥. So x and y are both limit
points of hfi ↑ {x} u hfi ↑ {y}, and thus f is not Hausdorff.
2⇒1: Suppose F is proper.
hfi ↑ {x} w F ∧ hfi ↑ {y} w F ⇒
hfi ↑ {x} u hfi ↑ {y} 6= ⊥ ⇒ x = y.
Corollary 36. Every entirely defined Hausdorff funcoid is Kol-
mogorov.
Remark 37. It is enough to be “almost entirely defined” (having
nonempty value everywhere except of one point).
Obvious 38. For a complete funcoid induced by a topological space
this coincides with the traditional definition of a Hausdorff topological
space.
7. Generalized limit vs axiomatic generalized limit
I will call singularities the set of generalized limits of the form
xlim
hµi↑{x}
f where f is an entirely defined funcoid and x ranges all
points of Ob µ.
Switching back and forth between generalized limits and what I call
F -singularities:
Φf =
(dom F, F )
F ∈ f
;
Ψf = im f.
Proposition 39. Let the funcoid µ is Kolmogorov and X is entirely
defined. Then:
(1) Φ is an injection from the set of singularities to the set of
monovalued functions.
(2) (f, X ) 7→ hΦ(xlim
X
f)i is an axiomatic generalized limit.
Proof. That it’s an injection is obvious.
We need to prove that dom F
0
6= dom F
1
for each F
0
, F
1
∈ f such
that F
0
6= F
1
. Really, F
0
= X ◦ f|
hµi↑{x
0
}
◦ ↑ r
0
for x
0
∈ Ob µ, r
0
∈ G.
8. OPERATIONS ON GENERALIZED LIMITS 17
We have dom F
0
= dom f|
hµi↑{x
0
}
= hµi ↑ {x
0
}. Similarly dom F
1
=
hµi ↑ {x
1
} for some x
1
∈ Ob µ. Thus dom F
0
6= dom F
1
because
otherwise x
0
= x
1
and so r
0
6= r
1
,
dom F
0
=
↑ r
−1
0
hµi ↑ {x
0
} =
hµi
↑ r
−1
0
↑ {x
0
} 6=
(Kolmogorov property) 6=
hµi
↑ r
−1
1
{x
0
} =
↑ r
−1
1
hµi {x
0
} = dom F
1
,
contradiction.
It remains to prove that (f, X ) 7→ Φ(xlim
X
f) conforms to the ax-
ioms.
The second axiom is obvious.
It remains to prove that
hΦ(xlim
X
f)i C = lim
C
f.
Really, Φ(xlim
X
f) is equal to an F ∈ xlim
X
f such that dom F = X .
So F = X ◦ f.
hΦ(xlim
X
f)i C = hX ◦ fi C = hX ◦ f i C = lim
C
f.
So if we define a function on the set of functions whose values are
funcoids, we automatically define (as this injection preimage) a function
on the set of singularities. Let’s do it.
Let ϕ be a (possibly multivalued) multiargument function.
8. Operations on generalized limits
8.1. Applying functions to functions. As usually in calculus:
Let ϕ is an arbitrary multiargument function.
Definition 40. (ϕf)x = ϕ(λi ∈ D : f
i
x) for an indexed (by
dom ϕ) family f of functions of the same domain D to domains of
arguments of φ.
8.2. Applying functions to sets.
Definition 41. ϕX = hϕi
∗
Q
X for a family X of sets, where each
X
i
is an element of the domain of i-th element of ϕ.
Obvious 42. ϕ(λi ∈ D : {x
0
}) = {ϕx}.
18 CONTENTS
8.3. Applying functions to filters.
Definition 43. ϕx = hϕi
Q
RLD
X∈
Q
x
X for a family x of atomic filters.
Proposition 44. ϕ can be continued to a pointfree funcoid.
Proof. Need to prove (theorem 1650 in [1])
hϕi
RLD
Y
X∈
Q
a
X v
l
F
D
x 7→ hϕi
Q
RLD
X∈
Q
x
X
E
∗
atoms X
X ∈ up a
.
Really,
G
*
x 7→ hϕi
RLD
Y
X∈
Q
x
X
+
∗
atoms X =
hϕi
G
*
x 7→
RLD
Y
X∈
Q
x
X
+
∗
atoms X .
But by theorem 1875 in [1]:
G
*
x 7→
RLD
Y
X∈
Q
x
X
+
∗
atoms X =
RLD
Y
X∈
Q
X
X.
So,
F
D
x 7→
Q
RLD
X∈
Q
a
X
E
∗
atoms X w
Q
RLD
X∈
Q
a
X . Thus follows the the-
sis.
8.4. Applying functions to funcoids.
Definition 45. For a family f of funcoids having a common source
set and filter on this set X
(ϕf)X = ϕ (λi ∈ dom f : hf
i
i X ) .
Proposition 46. It is a component of a funcoid.
Proof. As composition of two components of pointfree funcoids:
ϕ(f
0
, . . . , f
n
) = ϕ ◦ (X 7→ (λi ∈ dom f : hf
i
i X )).
Note that X 7→ (λi ∈ dom f : hf
i
i X ) is a component of a pointfree
funcoid because
8. OPERATIONS ON GENERALIZED LIMITS 19
Y 6 (λi ∈ dom f : hf
i
i X ) ⇔
∃i ∈ dom f : Y
i
6 hf
i
i X ⇔
∃i ∈ dom f : X 6
f
−1
i
Y
i
⇔
X 6
λi ∈ dom f :
f
−1
i
Y
i
=
X 6
Y 7→
λi ∈ dom f :
f
−1
i
Y
i
Y.
Proposition 47. Applying to funcoids is consistent with applying
to functions.
Proof. Consider values on principal atomic filters.
8.5. Applying to axiomatic generalized limits.
Definition 48. Define applying finitary (multivalued) functions ϕ
to and indexed family x of axiomatic generalized limits of the same
domain D (and probably different destination sets) as
ϕx = hXi ◦ ϕ ◦ x.
(Here φ is considered as a function on filters, x is considered as a
function on indexed famililies of functions.)
Proposition 49. If X ◦ X w X and reflexive and X commutes with
ϕ in some argument k, then
ϕx = φ ◦ x.
Proof. ϕ ◦ x v ϕx because X is reflexive.
x
i
= hX ◦ f
i
i for some funcoid f
i
.
x
i
X = hXi hf
i
i X .
ϕ ◦ x =
C 7→ ϕ(λi ∈ dom x : hXi hf
i
i C) =
C 7→ hXi ◦ ϕ ◦ (λi ∈ dom x : hXi hf
i
i C) =
hXi ϕ(C 7→ λi ∈ dom x : hXi hf
i
i C) w
ϕ
C 7→ λi ∈ dom x :
hXi hf
i
i C if i 6= k
hXi hXi hf
i
i C if i v k
w
ϕ (C 7→ λi ∈ dom x : hXi hf
i
i C) =
hXi ◦ ϕ ◦ (C 7→ λi ∈ dom x : hXi hf
i
i C) =
C 7→ hXi ϕ (λi ∈ dom x : hXi hf
i
i C) = hXi ◦ φ ◦ x = ϕx.
20 CONTENTS
Proposition 50. The conditions of the previous propositions hold
for A and for L if they hold for d. More exactly:
Let d is transitive (d ◦ d v d) and reflexive. Then:
(1) Let A is transitive (A ◦ A v A) and reflexive.
(2) Let L is transitive (L ◦ L v L) and reflexive.
Proof.
1. Reflexivity is obvious. Prove that A ◦ A v A. Really, A ◦ A =
(c ◦ d
−1
) ◦ (c ◦ d
−1
) v c ◦ d
−1
◦ d
−1
v c ◦ d
−1
= A where c is the
“core part” funcoid.
2. Reflexivity is obvious. Prove that L ◦ L v L. Really,
hLi hLi A = hLi
x
hdi
∗
{x} w A
= hLi
x
hdi
∗
{x} 6 A
=
hLi
x
{x} 6 hd
−1
i A
v hLi
d
−1
A =
x
hdi
∗
{x} w hd
−1
i A
v
x
hdi
∗
{x} w A
= hLi A.
Definition 51. Applying to singularities: ϕx = Ψf(Φ ◦ x) (appli-
cable only if limits x
i
are taken on filters that are equal up to hri for
r ∈ G).
Theorem 52. If ϕ is continuous regarding X in each argument and
dom f
0
= · · · = dom f
n
= ∆ and X ◦ X v X, then for singularities
lim ϕ(f
0
, . . . , f
n
) = ϕ(lim f
0
, . . . , lim f
n
).
Proof. We will prove instead
ϕ(Φ lim f
0
, . . . , Φ lim f
n
) = Φ lim ϕ(f
0
, . . . , f
n
).
Equivalently transforming:
λ∆ ∈ D : hXi ϕ((Φ lim f
0
)∆, . . . , (Φ lim f
n
)∆) =
Φ lim ϕ(f
0
, . . . , f
n
);
X ◦ ϕ(X ◦ f
0
◦ r, . . . , X ◦ f
n
◦ r) = X ◦ ϕ(f
0
, . . . , f
n
) ◦ r;
X ◦ ϕ(X ◦ f
0
, . . . , X ◦ f
n
) = X ◦ ϕ(f
0
, . . . , f
n
);
Obviously, X ◦ ϕ(X ◦ f
0
, . . . , X ◦ f
n
) w X ◦ ϕ(f
0
, . . . , f
n
).
8. OPERATIONS ON GENERALIZED LIMITS 21
Reversely, applying continuity n + 1 times, we get:
X ◦ ϕ(X ◦ f
0
, . . . , X ◦ f
n
) v
X ◦ X
|{z}
n+1 times
◦ ϕ(f
0
, . . . , f
n
) v
X ◦ ϕ(f
0
, . . . , f
n
).
So X ◦ ϕ(X ◦ f
0
, . . . , X ◦ f
n
) = X ◦ ϕ(f
0
, . . . , f
n
).
Proposition 53. If φ is continuous regarding X in each argument,
then
ϕ(lim f
0
|
∆
, . . . , lim f
n
|
∆
) =
lim ϕ(f
0
|
∆
, . . . , f
n
|
∆
) =
lim
∆
ϕ(f
0
, . . . , f
n
)
for funcoids f
0
, ..., f
n
,
Proof. The first equality follows from the above.
It remains to prove
ϕ(f
0
|
∆
, . . . , f
n
|
∆
) = (ϕ(f
0
, . . . , f
n
))|
∆
.
Equivalently transforming,
hϕ(f
0
|
∆
, . . . , f
n
|
∆
)i X = h(ϕ(f
0
, . . . , f
n
))|
∆
i X ;
ϕ (hf
0
|
∆
i X , . . . , hf
n
|
∆
i X ) =
hϕ(f
0
, . . . , f
n
)i (∆ u X );
ϕ (hf
0
i (∆ u X ), . . . , hf
n
i (∆ u X )) =
ϕ (hf
0
i (∆ u X ), . . . , hf
n
i (∆ u X )) (∆ u X ).
Theorem 54. Let ∆ be a filter on µ. Let S be the set of all
functions p ∈ FCD(Ob µ, Ob X) such that dom p = ∆. Let f, g be
finitary multiargument functions on Ob X. Let J be an index set. Let
k ∈ J
dom P
, l ∈ J
dom Q
. Then
∀x ∈ (Ob X)
J
: f(λi ∈ dom f : x
k
i
) =
g(λi ∈ dom g : x
l
i
)
22 CONTENTS
implies
∀x ∈ (hlimi
∗
S)
J
: f(λi ∈ dom f : x
k
i
) =
g(λi ∈ dom g : x
l
i
),
provided that f and g are continuous regarding X in each argument.
Remark 55. This theorem implies that if Ob X is a group, ring,
vector space, etc., then hlimi
∗
S is also accordingly a group, ring, vector
space, etc.
Proof. Every x
j
i
= lim
∆
t for some function t.
By proved above,
f(λi ∈ dom f : x
k
i
) = lim
∆
f(λi ∈ dom f : t
k
i
).
It’s enough to prove
f(λi ∈ dom f : t
k
i
) = g(λi ∈ dom f : t
l
i
).
But that’s trivial.
Conjecture 56. The above theorem stays true if S is instead a
set of limits of monovalued funcoids.
8.6. Applications. Having generalized limit, we can in an obvious
way define derivative of an arbitrary function.
We can also define definite integral of an arbitrary function (I re-
mind that integral is just a limit on a certain filter). The result may
differ dependently on whether we use Riemann and Lebesgue integrals.
From above it follows that my generalized derivatives and integrals
are linear operators.
9. Equivalence of different generalized limits
Proposition 57. Axiomatic generalized limits of monovalued fun-
coids for X = A and X = L coincide on ultrafilers.
Proof. Follow from the facts that the image of an ultrafilter by an
atomic funcoid is an ultrafilter and that L and A coincide on ultrafilters.
Question 58. Under which conditions the algebras of all functions
on the set of axiomatic singularities between two fixed sets A and B
induced (as described above) by functions A → B for generalized limits
are pairwise isomorphic (with an obvious bijection) for:
(1) X = A;
(2) X = L.
10. HIERARCHY OF SINGULARITIES 23
Moreover, if d is T
2
-separable, they are also isomorphic to the case
X = d ◦ d
−1
(remark: for a pretopology d, it’s the proximity of two
sets being near if they have intersecting closures). The isomorphism is
composing every element F of an axiomatic singularity with d on the
left (F 7→ d ◦ F ).
Also let question how to generalize the above for functions φ be-
tween different kinds of singularities is also not yet settled.
10. Hierarchy of singularities
Above we have defined (having fixed endofuncoids µ and X) for
every set of “points” R = Ob X its set of singularities SNG(R).
We can further consider
SNG(SNG(R)), SNG(SNG(SNG(R))),
etc.
If we try to put our generalized derivative into say the differential
equation h◦f
0
= g◦f on real numbers, we have a trouble: The left part
belongs to the set of functions to SNG(Y ) and the right part to the set
of functions to Y , where Y is the set of solutions. How to equate them?
If Y would be just R we would take the left part of the type SNG(R)
and equate them using the injection τ defined above. But stop, it does
not work: if the left part is of SNG(R) then the right part, too. So the
left part would be SNG(SNG(R)), etc. infinitely.
So we need to consider the entire set (supersingularities)
SUPER(R) = R ∪ SNG(R) ∪ SNG(SNG(R)) ∪ . . .
But what is the limit (and derivative) on this set? And how to
perform addition, subtraction, multiplication, division, etc. on this set?
Finitary functions on the set SUPER(R) are easy: just apply τ to
arguments belonging to “lower” parts of the hierarchy of singularities a
finite number of times, to make them to belong to the same singularity
level (the biggest singularity level of all arguments).
Instead of generalized limit, we will use “regular” limit but on the
set SUPER(R) (which below we will make into a funcoid) rather than
on the set R.
See? We have a definition of (finite) differential equations (even
partial differential equations) for discontinuous functions. It is just a
differential equation on the ring SUPER(R) (if R is a ring).
What nondifferentiable solutions of such equations do look like? No
idea! Do they contain singularities of higher levels of the above hierar-
chy? What about singularities in our sense at the center of a blackhole
24 CONTENTS
(that contain “lost” information)? We have something intriguing to
research.
11. Funcoid of singularities
I remind that for funcoid X the relation [X]
∗
can be thought as
generalized nearness.
We will extend [X]
∗
from the set R of points to the set of funcoids
from a (fixed) set A to R having the same domain (or empty domain):
y
0
[X]
∗
y
1
⇔ ∀x ∈ atoms dom y
0
: hy
0
i x [X]
∗
hy
1
i x
where atoms dom y
0
is the set of ultrafilters over the filter dom y
0
.
The above makes X a pointfree funcoid (as defined in [1]) on this
set of funcoids:
Proof. Because funcoids are isomorphic to filters on certain boolean
lattice, it’s enough to prove:
¬(⊥ [X]
∗
y
1
), ¬(y
0
[X]
∗
⊥),
i t j [X]
∗
y
1
⇔ i [X]
∗
y
1
∨ j [X]
∗
y
1
,
y
0
[X]
∗
i t j ⇔ y
0
[X]
∗
i ∨ y
0
[X]
∗
j.
The first two formulas are obvious. Let’s prove the third (the fourth is
similar):
i t j [X]
∗
y
1
⇔
∀x ∈ atoms dom(i t j) : hy
0
i x [X]
∗
hy
1
i x ⇔
∀x ∈ atoms(dom i t dom j) : hy
0
i x [X]
∗
hy
1
i x ⇔
∀x ∈ atoms dom i ∪ atoms dom j :
hy
0
i x [X]
∗
hy
1
i x ⇔
∀x ∈ atoms dom i : hy
0
i x [X]
∗
hy
1
i x∨
∀x ∈ atoms dom j : hy
0
i x [X]
∗
hy
1
i x ⇔
i [X]
∗
y
1
∨ j [X]
∗
y
1
.
We will define two singularities being “near” in terms of F -singularities
(that are essentially the same as singularities):
Two F -singularities y
0
, y
1
are near iff there exist two elements of y
0
and y
1
correspondingly such that dom y
0
= dom y
1
and every Y
0
∈ y
0
,
Y
1
∈ y
1
are near.
Let’s prove it defines a funcoid on the set of F -singularities:
12. FUNCOID OF SUPERSINGULARITIES 25
Proof. Not ∅ [X]
∗
X and not X [X]
∗
∅ are obvious.
It remains to prove for example
I ∪ J [X]
∗
K ⇔ I [X]
∗
K ∨ J [X]
∗
K,
but that’s obvious.
12. Funcoid of supersingularities
It remains to define the funcoid of supersingularities.
Let y
0
, y
1
be sets of supersingularities.
We will define y
0
and y
1
to be near iff there exist natural n, m such
that
τ
n
[y
0
∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)].
Remark 59. In this formula both the left and the right arguments
of [X]
∗
belong to SNG
n+m
(R).
Let’s prove that the above formula really defines a funcoid:
Proof. We need to show
∅ [X]
∗
y
1
, y
0
[X]
∗
∅,
i ∪ j [X]
∗
y
1
⇔ i [X]
∗
y
1
∨ j [X]
∗
y
1
,
y
0
[X]
∗
i ∪ j ⇔ y
0
[X]
∗
i ∨ y
0
[X]
∗
j.
26 CONTENTS
The first two formulas are obvious. Let’s prove the third (the fourth is
similar):
i ∪ j [X]
∗
y
1
⇔
∃n, m ∈ N :
τ
n
[(i ∪ j) ∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)] ⇔
∃n, m ∈ N :
τ
n
[(i ∩ SNG
m
(R)) ∪ (j ∩ SNG
m
(R)))] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)] ⇔
∃n, m ∈ N :
τ
n
[i ∩ SNG
m
(R)] ∪ τ
n
[j ∩ SNG
m
(R)))] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)] ⇔
∃n, m ∈ N :
(τ
n
[i ∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)]∨
τ
n
[j ∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)]) ⇔
∃n, m ∈ N :
τ
n
[i ∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)]∨
∃n, m ∈ N :
τ
n
[j ∩ SNG
m
(R)] [X]
∗
τ
m
[y
1
∩ SNG
n
(R)] ⇔
i [X]
∗
y
1
∨ j [X]
∗
y
1
.
13. Example differential equation
Definition 60. I will call a function f pseudocontinuous on D
when
∀a ∈ D : xlim
∆{a}\{a}
f = f(a).
13.1. Arbitrary pseudocontinuous continuations. Note that
arbitrary pseudocontinuous continuations of generalized solutions of
differential equations (diffeqs) are silly:
Let A(f(x), f
0
(x)) = 0 is a diffeq and let the equality is undefined at
some point (e.g. contains division by zero). Let f be its solution with
derivative f
0
. Replace the value in undefined point x of the solution by
an arbitrary value y and calculate the derivative y
0
at this point. No
need to hold A(y, y
0
) = 0 at this point because the point is outside of the
domain of the original solution. Then replace in our solution f the value
at this point x by y and the derivative by y
0
. Then we have another
13. EXAMPLE DIFFERENTIAL EQUATION 27
continuation of the solution because the equality A(f(x), f
0
(x)) = 0
holds both for the point x and all other points.
Thus, we can take any solution and add one point of it with an
arbitrary value. That’s largely a nonsense from the practical point of
view. (Why we would arbitrarily change one point of the solution?)
13.2. Solutions with pseudocontinuous derivative. So I will
require for generalized solutions instead that the derivative f
0
is pseu-
docontinuous.
Next, we will consider a particular example, the diffeq y
0
(x) =
−1/x
2
. Let us find its continuations of generalized solutions y to the
entire real line (including x = 0) with a y
0
being pseudocontinuous.
As it’s well known, its solutions in the traditional sense are y(x) =
c
1
+
1
x
for x < 0 and y(x) = c
2
+
1
x
for x > 0 where c
1
, c
2
are arbitrary
constants. The derivative is y
0
(x) = −1/x
2
.
Remark 61. We could consider solutions on the space of supersin-
gularities and it would be the same, except that we would be allowed
to take c
1
, c
2
arbitrary supersingularities instead of real numbers. This
is because the supersingularities form a ring and thus the algorithm of
solving the diffeq is the same as for the real numbers, thus producing
the solutions of the same form.
Let’s find the pseudocontinuous generalized derivative at zero by
pseudocontinuity:
y
0
(0) = lim
x→0
1
x
2
.
On the other hand, by the definition of derivative
y
0
(0) = lim
ε→0
c
i
+
1
ε
− f(0)
ε
.
The equality is possible only when c
1
= c
2
= c = f(0).
So, finally, our solution is y(x) = c +
1
x
for x 6= 0 and y(0) = c.
A thing to notice that now the solution is “whole”: it exists at zero
and does not split to two “branches” with independent constants. Our
y(x) is a real function, but the derivative has a singularity in my sense.
We considered generalized solutions with pseudocontinuous deriv-
ative. It is apparently the right way to define a class of generalized
solutions. Now I will consider also several apparently wrong classes of
solutions.
13.3. Pseudocontinuous generalized solutions. Let us try to
require the solution of our diffeq to be pseudocontinuous instead of its
derivative to be pseudocontinuous.
28 CONTENTS
We already have the solution for nonzero points. For zero:
y
0
(0) = lim
x→0
(−1/x
2
).
So the derivative:
y
0
(0) = lim
x→0
y(x) − y(0)
x
=
lim
x→0
y(x)
x
− lim
x→0
y(0)
x
=
lim
x→0
1
x
2
− lim
x→0
y(0)
x
.
The equality is impossible.
