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This is an exercise from Rudin's Real and Complex Analysis book.

Construct a monotonic function $f$ on $\Bbb R$ so that $f'(x)$ exists (finitely) for every $x\in \Bbb R$ but $f'$ is not a continuous function.

How can I construct such a function?

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    What does this question have to do with measure theory? – 2017-02-26
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    Discontinuous in how many points? – 2017-02-26
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    @Omnomnomnom Maybe the function is almost an integral or something like that? I don't know. It is in a measure theory book. – 2017-02-26
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    @YiorgosS.Smyrlis At least one, I think – 2017-02-26
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    @YiorgosS.Smyrlis Well it has to be a finite amount otherwise $f^\prime$ wouldn't exist over every $x\in \mathbb R$. However one should work. – 2017-02-26
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    @Omnomnomnom: Maybe Rudin had some Cantor function shenanigans in mind? – 2017-02-26

2 Answers 2

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Example $$ f(x)=\left\{ \begin{array}{ccc} 2x & \text{if} & x\le 0,\\ 2x+x^2+x^2\sin(1/x) & \text{if} & x> 0. \end{array} \right. $$ Then $$ f'(x)=\left\{ \begin{array}{ccc} 2 & \text{if} & x\le 0,\\ 2+2x+2x\sin(1/x)-\sin(1/x) & \text{if} & x> 0. \end{array} \right. $$

Clearly $f$ is increasing and differentiable everywhere, but $f'$ is discontinuous at $x=0$.

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Start with

$$f(x)=\begin{cases} x^2\sin(1/x) & x\neq 0\\ 0 & x=0 \end{cases}$$

It is well known that

$$f'(x)=\begin{cases} 2x\sin(1/x) -\cos(1/x) & x\neq 0 \\ 0 & x = 0 \end{cases} $$ is discontinous at $x=0$. However $f'(x)$ is bounded. So let $C$ be some lower bound of $f'$. Define $g(x)=f(x)+Cx$. Then $g'(x)=f'(x)+C > 0$ everywhere and thus by the first derivative test $g$ is increasing yet $g'$ is still discontinous at $x=0$.

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    I would like to accept your answer too, but Yiorgos had this idea first, although your answer is more simple. Thanks man +1 – 2017-02-27