# Familiarizing with the Fundamental Theorem of Calculus

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!

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.