7.5. The Basic Representation TheoremTesting the differentiability of a function usually means to look for a continuous extension of the difference quotient function, a sometimes cumbersome procedure but working with many functions. There are however seriuous difficulties with functions like sin, cos or exp, i.e. functions too important to ignore them. Fortunately we will succeed in this and similar cases using an equivalent criterion, the basic representation theorem, to be introduced in this chapter. This theorem will also significantly bring foward our subject.
Consider:
Continuity and differentiability are two basic concepts with functions. The representation theorem reveals how they are related.
The absolute value function is continuous everywhere and fails to be differentiable at only one point, which is of course sufficient for a counter example to [7.5.2]. It is quite a challenge to look for continuous functions that are nowhere differentiable. The first to construct a function like that was Karl Weierstrass in 1872. The just established continuity of a differentiable function is not the only property resulting from the representation theorem [7.5.2]. There are many more, and as another example we will show that the inverse of an injective function f is differentiable as well if f is regular at a, i.e. if ${f}^{\prime}(a)\ne 0$.
Later on we will see that functions being regular on an entire interval will be injective automatically. This is no longer true, not even locally, with only pointwise regular functions. Click for an example. We will employ the representation theorem again to get more results on differntiable functions within the next chapters. Now we use the theorem to settle the differentiability for an important class of functions.
[7.5.5] guarantees the differentiability for a wide class of functions: An analytical function $f:A\to \mathbb{R}$ coincides at each $a\in A$ with the limit function of a convergent power series expanded at a. Thus we have:
Using the rearrangement theorem for convergent power series will yield a considerable extension of [7.5.5]: Limit functions of convergent power series are differentiable on the whole of their domain of convergence. The derivative is calculated with an easy scheme!
exp, sin and cos are analytical functions (c.f. [5.12.4]) and thus differentiable at each $b\in \mathbb{R}$. Recalling that
$\begin{array}{l}\mathrm{exp}=\sum _{i=0}^{\infty}\frac{1}{i!}{\mathrm{X}}^{i}\hfill \\ \mathrm{sin}=\sum _{i=0}^{\infty}\frac{{(1)}^{i}}{(2i+1)!}{\mathrm{X}}^{2i+1}\hfill \\ \mathrm{cos}=\sum _{i=0}^{\infty}\frac{{(1)}^{i}}{(2i)!}{\mathrm{X}}^{2i}\hfill \end{array}$
(see [5.11.12] for details) we may calculate their derivations using [7.5.7]. Note that the value of the first addend in [7.5.7] equals to $a}_{1}{(ba)}^{0$. To consider this for the sine we set the initial value $i=0$ when proving 2. With the Cosine we have ${a}_{1}=0$ so that the derivation in 3. correctly starts with $a}_{2}{(ba)}^{1$.
