The range of the derivative of a differentiable function

Question

I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be

(1) a closed interval or
(2) an open interval or
(3) a half-open interval or
(4) an unbounded interval

Can someone give an example for each one ?

Answer 1

Let's see: For (1), any $f$ with continuous derivative works. The typical example for (4) is the continuous extension of $x^2\sin(1/x^2)$ to $[0,1]$.

An example for (2) is a bit harder. This comes from

Bernard R. Gelbaum, John M. H. Olmsted. Counterexamples in Analysis, Dover Books on Mathematics, 2003.

Differentiation is discussed in chapter 3. Consider the function $$ f(x)=\left\{\begin{array}{cc}x^4e^{-x^2/4}\sin(8/x^3)&x\ne0,\\ 0&x=0,\end{array}\right. $$ for $-1\le x\le 1$. Its derivative is $$ f'(x)=\left\{\begin{array}{cc}e^{-x^2/4}\left[\bigl(4x^3-\frac12x^5\bigr)\sin(8/x^3)-24\cos(8/x^3)\right]&x\ne0,\\ 0&x=0,\end{array}\right. $$ which has range $(-24,24)$ (in any neighborhood of zero). A proof is in their book (pp. 37-38). The reference they provide is

John M. H. Olmsted. Advanced calculus, Appleton Century Crofts, Inc., New York, 1961.

I include the graph of $f'$ produced by WolframAlpha:

Once we have an example for (2), an example for (3) is easy: Simply pick some $t\ne0$ where $f'(t)=0$, say $0<t<1$, and replace $f$ to the right of $t$ with a continuous function $g$ such that $g(t)=f(t)$, $g'(t)=0$, and $g'$ has range $[-24,0]$.