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Consider the following form of the Fundamental Theorem of Calculus: "Let $f:[a,b] \rightarrow \mathbb{R}$ be a differentiable function. Suppose that $F'$ is Riemann integrable over $[a,b]$. Then $\int^{b}_{a}F'=F(b)-F(a)$."

In order to familiarize with this theorem, it must be remarked that there is no condition for the integrand to be continuous. Now I am looking for a concrete example of a discontinuous, but integrable integrand $f$ which is the derivative of a differentiable function $F$.

I came up with $f:[0,2]\rightarrow \mathbb{R}:x \mapsto\begin{cases}1 \text{ if } 0\leq x \leq 1 \\ 2 \text{ if }1<x \leq2\end{cases}$. This integrand is discontinuous, but it is the derivative of a function which is not everywhere differentiable on $[0,2]$ (i.e. on $1$). Can someone give a better example? Thank you!

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The classic example is the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=x^2 \sin(1/x)$ for $x \neq 0$ and $f(0)=0$. The function is differentiable everywhere except at 0. Of course by shifting the function we can now get functions whose derivatives are continuous everywhere except one point.

Some more interesting examples are given in this answer: Discontinuous derivative.

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