A REMARK ON THE DIFFERENTIABILITY OF GERMS

Edward Tutaj

IMUJ Preprint 2000/26

Abstract

The notion of continuous differentiability for functions from R to to R is described in a somewhat non-typical language.

The remark presented in this paper is - if one looks from purely mathematical point of view - very simple. We present it, since this remark concerns something, which is so fundamental as the definition of differentiability. We will restrict ourselves to the simplest possible situation, i.e. to the differentiation of functions from R to R, although some more general observations are also true.

1 Quasialgebras

First we will adapt some known notions to a slightly new situation. Let us recall, that an algebra A (over R) is an algebraic structure with three operations: addition, multiplication and scalar multiplication. Since it occurs, that the multiplication is not commutative, then to have an algebra one must assume two distribution laws. Namely

(i) : a · (b + c) = a · b + a · c

and

(ii) : (b + c) · a = b · a + c · a.

((i) will be said the left and (ii) the right distribution law).

Definition 1.1. A quasialgebra is an algebraic structure A, which satisfies all the axioms needed in the case of a non-commuta-tive algebra with unit, without perhaps the left distribution law.

Clearly, each algebra is a quasialgebra. Some essential examples of quasialgebras will be presented below. We introduce this nontypical notion only to have the possibility of using the language used in the case of ”true” algebras. In particular:

a). It is clear what one will mean by ”subquasialgebra” of a given quasialgebra;

b). It is clear what one will mean by ”homomorphism of quasialgebras”;

c). Since we have a unit - say e - in A, we may define the regular elements in A. Namely

Definition 1.2. An element x ∈ A will be said to be regular if there exists an element y ∈ A such that x · y = y · x = e.

If x is not regular, it will be said to be singular.

d). Simiraly, as in the case of algebras, we have

Definition 1.3. If x ∈ A then the spectrum of x is the set

Sp(x) = {λ ∈ R : x − λ · e is singular}.

Clearly,the elements of Sp(x) will be said the eigenvalues of x.

2 Different spaces of germs

First - for establish the notations - we will recall some standard definitions. Let F denote the family of all real functions defined in a neighborhood of 0 in R. If f ∈ F, g ∈ F then the relation :

f ∼ g ⟺ ∃V : V is a neighbourhood of 0 and f|_{V} = g|_{V}

is an equivalence relation in F.

The quotient F/ ∼ will be denoted by F̂ and the equivalence class of f ∈ F will be denoted by f̂ and called the germ of f.

There is a natural way to introduce the addition and the multiplication to F̂, namely

F̂ = limind_{V →{0}}F(V),

where V is a neighbourhood of the origin and

F(V) = {f : V → R}

is the algebra of real functions on V with natural addition and multiplication.

We will say that a germ f̂ is bounded if there exists a bounded function g such that f ∼ g. Similarly, we will say that a germ f̂ is continuous (resp. differentiable) it there exists a continuous (resp. differentiable) at 0 function g, such that g ∼ f. Finally, we will say that that a germ f̂ is lipschitzian if there exist a function g satisfying the Lipschitz condition and equivalent to f. The subspaces (of F̂) of bounded, continuous, differentiable, or lipschitian germs will be denoted by Ô, Ĉ, D̂ and L̂ respectively.

The value of a germ f̂ at 0 (denoted by f̂(0)) is by definition the common value at 0 of all functions equivalent to f. If a germ f̂ is differentiable then in an analogous way one defines the value f̂′(0).

Let

Ĉ_{o} = {f̂ ∈ Ĉ: f̂(0) = 0}.

We define D̂_{o} and L̂_{o} as the suitable subspaces of Ĉ.

Let us look now at the space Ĉ_{o}. As a subspace of Ĉ with standard multiplication, this is an ideal, but for our purposes it is essential to observe that one can define the composition of germs in Ĉ_{o}. Indeed, let f̂ and ĝ be two germs from Ĉ_{o} and let g_{1} : W → R and f_{1} be the functions equivalent to g and f respectively. Since f_{1} is continuous then there exists a neighbourhood V, such that f_{1}(V) ⊂ W. Hence one can compose f_{1}|V with g_{1} and the germ of this composition (i.e. ĝ_{1} ∘ f_{1}) is by definition the composition of germs ĝ∘f̂. One can easily check that this definition is independent on the choice of f_{1} and g_{1}.

Consider now the set Ĉ_{o} together with the operations of addition, scalar multiplication and compositions of germs. With respect to the addition and scalar multiplication Ĉ_{o} is a vector space. The operation of composition of germs is clearly associative, non commutative, but with the neutral element î (î the germ of the identity). However we cannot state that (Ĉ_{o}, +, ∘, ·) is an algebra, because the distribution law does not hold. More exactly, since the composition of germs is not commutative, we must consider left, and right distribution laws, namely:

(f̂+ ĝ) ∘ ĥ = f̂∘ ĥ + ĝ ∘ ĥ

and

ĥ ∘ (f̂+ ĝ) = ĥ ∘ f̂+ ĥ ∘ ĝ.

It is easy to see, that the first of this properties is fullfiled (the right distribution property) while the second (the left distribution property) does not hold in general. Taking into account the definition 1.2. we may state that Ĉ_{o} is a quasialgebra with composition and that D̂_{o} (and L̂_{o}) are the subquasialgebras of Ĉ_{o}, since the composition of differentiable (lipschitzian) germs is a differentiable (lipschitzian) germ.

3 A remark

In this section we will work with the quasialgebra L̂_{o}. For a germ f̂ ∈ L̂_{o} we set (f is an element of of f̂)

I_{f}(x, y) = \frac{f(x) − f(y)}{x − y}.

Let us observe the following

Lemma 3.1. For each f̂ ∈ L̂_{o} we have: f̂ is singular if and only if there exist sequences x_{n} → 0, y_{n} → 0, x_{n} ≠ y_{n} such that

I_{f}(x_{n}, y_{n}) → 0.

Proof: Suppose that f̂ is regular. This means that there exist a neighbourhoods of 0 - say V and W - and the Lipschitz functions f : V → W (f is an element of f̂) and g : W → V , which are inverse to each other. Clearly we may assume that x_{n} ∈ V, y_{n} ∈ V and then there exist u_{n} ∈ W, v_{n} ∈ W such that g(u_{n}) = x_{n} and g(v_{n}) = y_{n}.

Hence, by the definition of I_{f} and from our assumption we obtain

\frac{u_{n} − v_{n}}{g(u_{n}) − g(v_{n})} → 0,

but this is impossible since g satisfy the Lipschitz condition.

To prove the converse, let us observe that if U is a connected neighbourhood of 0 then

I_{f}(U × U \ {(x, x) : x ∈ U})

is connected in R. Hence we may suppose (without loss of generality) that there exist a neighbourhood U of 0 and and a real δ > 0 such that

\frac{f(x) − f(y)}{x − y} > δ,

for all x ∈ U, y ∈ U, x ≠ y. But this means that f is strictly increasing in U and then has an inverse g : W → U for a moment in Ĉ_{o}. But from the inequality above it follows that

0 < g(u) − g(v) < δ^{−1} · (u − v),

for all u ∈ W, v ∈ W, u ≠ v, which means that the germ ĝ is the inverse of f̂ in L̂_{o}. This ends the proof of 3.1.

Lemma 3.2:Let us set for f̂ ∈ L̂_{o}

a(f̂) = lim inf_{x→0,y→0,x≠y} \frac{f(x) − f(y)}{x − y}

and

b(f̂) = lim sup_{x→0,y→0,x≠y} \frac{f(x) − f(y)}{x − y}.

Then Sp(f̂) = [a(f̂), b(f̂)].

Proof. It follows from the remark on the connectedness made above, that for each point from [a(f̂), b(f̂)] - say α - there exist sequences x_{n} → 0, y_{n} → 0, x_{n} ≠ y_{n} such that I_{f}(x_{n}, y_{n}) → α. But by 3.1. this means that the germ of the function x → f(x) − α · x is singular.

On the other hand if c < a(f̂) (when b(f̂) < c the proof is similar) and δ = 2^{−1}(a(f̂)−c) then there exists a neighbourhood U of 0 such that

I_{f}(x, y) − a(f̂) > δ,

for all x ∈ U, y ∈ U. But this means that the germ of the function x → f(x) − c · x is regular in L̂_{o}. This ends the proof of 3.2.

Now we can restate 3.2. in a following form

Theorem 3.3. If a germ f̂ ∈ L̂_{o} has only one eigenvalue, then it is differentiable.

Proof.It follows directly from 3.2.

Example 3.4. The converse in 3.3. does not hold. The function f(x) = x^{2} · sin \frac{1}{x} (and f(0) = 0) is differentiable , its derivative is bounded (then f is lipschitzian) and the spectrum of the germ of f is [−1, 1].

4 Further remarks

We will come back now to the notion of spectrum of an element. For t ∈ R we denote by τ_{t} the translation R ∋ x → x + t ∈ R. If f : (a, b) → R is a function satisfing locally the Lipschitz condition and x ∈ (a, b) then we set:

f̂_{x} = ˆ (τ_{−f(x)} ∘ f ∘ τ_{x}).

Clearly, for each x ∈ (a, b) the germ f̂_{x} is in L̂_{o}. It is not hard to observe that the spectrum of each germ in L̂_{o} is not empty, but in general this spectrum may have more than one element.

Definition 4.1. We will say that a function f : (a, b) → R is S-differentiable if it satisfies locally the Lipschitz condition and for each x ∈ (a, b) the spectrum of the germ f̂_{x} (in L̂_{o}) has exactly one element.

This definition is justified by the following

Theorem 4.2. A function f : (a, b) → R is S-differentiable if and only if is of class C_{1} on (a, b).

Proof: Suppose that f ∈ C_{1}(a, b). Then f is locally lipschitzian. Let us fix x_{o} ∈ (a, b). We will show that Sp(f̂_{x_{o}}) = {f′(x_{o})}. Indeed, let

ĝ_{x_{o}} = f̂_{x_{o}} − f′(x_{o}) · î.

Clearly, g′_{x_{o}}(0) = 0 and then by 3.1. ĝ_{x_{o}} is singular, which means that f′(x_{o}) ∈ sp(f̂_{x_{o}}).

Moreover, for a real δ ≠ 0 we set ĥ_{δ} = f̂_{x_{o}} − (f′(x_{o}) + δ) · î. Then for some function h_{δ} we have:

h_{δ}(x) = (f(x + x_{o}) − f(x_{o}) − f′(x_{o}) · x) − δ · x

and we see that h_{δ} is C_{1} in a neighbourhood of x_{o} and h′_{δ}(0) = −δ. But this means that ĥ_{δ} is regular in L̂_{o} and then f′(x_{o})+δ ∉ Sp(f̂_{x_{o}}).

To prove the converse, suppose that spectrum f̂_{x_{o}} ={a}. Consider the germ of the function:g(x) = f(x + x_{o}) − f(x_{o}) − a · x. This germ satisfies 3.1. hence g′(0) = 0 and in consequence f′(x_{o}) = a. To prove the continuity of f′ it is sufficient to show that

lim_{x→x_{o},y→x_{o},x≠y} \frac{f(x) − f(y)}{x − y} = f′(x_{o}).

It is clear that if this limit exists, it equals f′(x_{o}), hence suppose that there exist the sequences {x_{n}}_{1}^{∞}, {y_{n}}_{1}^{∞}, {u_{n}}_{1}^{∞}, {v_{n}}_{1}^{∞} and the reals b and c, such that b ≠ c ,

lim_{n→ infty} x_{n} = lim_{n→∞} y_{n} = lim_{n→∞} u_{n} = lim_{n→∞} v_{n} = x_{o}

and

lim_{n→∞} \frac{f(x_{n}) − f(y_{n})}{x_{n} − y_{n}} = b

and

lim_{n→∞} \frac{f(u_{n}) − f(v_{n})}{u_{n} − v_{n}} = c.

Consider now the germ of the function

h(x) = f(x + x_{o}) − f(x_{o}) − b · x.

Setting x̃_{n} = x_{n} − x_{o} and ỹ_{n} = y_{n} − x_{o} we obtain

h(x̃_{n}) − h(ỹ_{n}) = f(x_{n}) − f(y_{n}) − b · (x_{n} − y_{n}).

Hence

\frac{h(x̃_{n}) − h(ỹ_{n})}{x̃_{n} − ỹ_{n}} = \frac{f(x_{n}) − f(y_{n})}{x_{n} − y_{n}} − b

since x̃_{n} − ỹ_{n} = x_{n} − y_{n}.

Thus the derivative of h at 0 is 0, and in consequence h is singular. But this means that b is in the spectrum of f̂_{x_{o}}. Since the same is true for c we obtain a contradiction with our assumption on the spectrum of the germ f̂_{x_{o}}.

5 A comment

Roughly speaking the differentiation of germs is a projection from an quasialgebra of germs (exactly those which are differentiable) onto the (quasi)algebra of germs of linear functions, and this projection is also a homomorphism of quasialgebras. Our theorem shows that at least in L̂_{o} the subquasialgebra D̂_{o} ∩ L̂_{o} is uniquely determined by these two conditions i.e. ”projection” and ”homomorphism”.