The Derivative as a Function

Section 6.3 The Derivative as a Function

For the rest of the chapter, let's specialize to functions \(\mathbb{R}\to\mathbb{R}\text{;}\) the nitty gritty of extending the theorems in we state here to higher-dimensional normed spaces is more properly the subject of a second course in real analysis (such as MATH 3610).

Definition 6.3.1.

If we regard the assignment \(c\mapsto Df(c)\) as a function, that's totally fine to do.

In particular, we can formulate questions about the function \(f'\text{.}\)

Definition 6.3.2.

If the function \(f':x\mapsto f'(x)\) is continuous, then we call \(f\) continuously differentiable. The set of all continuously differentiable functions with domain \(A\) is denoted \(\mathcal{C}^1(A)\text{.}\)

Remark 6.3.3.

In order for "continuously differentiable" to be a sensible notion, we should convince ourselves that not every derivative is continuous.

Example 6.3.4.

Consider the function \(g:\mathbb{R}\to\mathbb{R}\) given by \(g(x)=\begin{cases} x\sin\left(\frac{1}{x}\right)& \text{ if }x\neq 0\\0&\text{ if }x=0\end{cases}\text{.}\) Then \(g\) is continuous everywhere but not differentiable at \(x=0\text{.}\)

Consider the function \(h:\mathbb{R}\to \mathbb{R}\) given by \(h(x)=\begin{cases}x^2\sin\left(\frac{1}{x}\right)& \text{ if }x\neq 0\\0&\text{ if }x=0\end{cases}\text{.}\) Then \(h\) is differentiable everywhere but \(\displaystyle\lim_{x\to 0}h'(x)\) does not exist, so \(h\notin\mathcal{C}^1(\mathbb{R})\text{.}\)

The weird behavior in Example 6.3.4 is oscillatory, rather than a jump discontinuity. This isn't an accident.

Theorem 6.3.5. Darboux's Theorem.

Let \(f:\mathbb{R}\to\mathbb{R}\) be a differentiable function and \(a,b\in \mathbb{R}\text{.}\) If \(z\) is any number between \(f'(a)\) and \(f'(b)\text{,}\) then there is \(c\in [a,b]\) with \(f'(c)=z\text{.}\)

Remark 6.3.6.

A function which satisfies the conclusion of Theorem 6.3.5 is called Darboux or said to have the intermediate value property.

So we might say that Bolzano's Intermediate Value Theorem says that continuous functions are Darboux and Darboux's Theorem says that derivatives are Darboux.

Definition 6.3.7.

The derivative of \(Df\) is called \(D^2f\text{;}\) its derivative is \(D^3f\text{;}\) etc.

It's not hard to expand Example 6.3.4 to get a function which has a second derivative which fails to be continuous, a function which has a third derivative which fails to be continuous, etc. So we have the following tower of sets of functions:

\begin{align*} \mathcal{C}^0(A)&=\left\{f\middle\vert f\text{ is continuous on }A\right\}\\ \mathcal{D}(A)&=\left\{f\middle\vert f\text{ is differentiable on }A\right\}\\ \mathcal{C}^1(A)&=\left\{f\middle\vert f\text{ is differentiable on }A\text{ and }f'\text{ is continuous on } A\right\}\\ \mathcal{D}^2(A)&=\left\{f\middle\vert f'\text{ is differentiable on }A\right\}\\ \mathcal{C}^2(A)&=\left\{f\middle\vert f'\text{ is differentiable on }A\text{ and }f''\text{ is continuous on } A\right\}\\ &\vdots \end{align*}

with

\begin{equation*} \cdots\subsetneq \mathcal{C^2}(A)\subsetneq\mathcal{D}^2(A)\subsetneq \mathcal{C}^1(A)\subsetneq \mathcal{D}(A)\subsetneq \mathcal{C}^0(A)\ \ . \end{equation*}

Checkpoint 6.3.8.

Using Example 6.3.4 as a template, show each inclusion in this tower is strict. That is, for each inclusion come up with a function which lies in the larger set but not in the smaller one.

Here's an important theorem, which we'll prove in class:

Theorem 6.3.9. Fermat's Theorem.

If \(f:[a,b]\to \mathbb{R}\) is differentiable on \((a,b)\) and \(c\in(a,b)\) is where \(f\) achieves an extremum, then \(f'(c)=0\text{.}\)

Checkpoint 6.3.10.

If \(f:[a,b]\to \mathbb{R}\) is differentiable and achieves its maximum or minimum at an endpoint, what can you say?