prove or find counter example to the statement : Riemann integralable

Question

If the function $f:[a, b]\rightarrow R $ is Riemann integralable on $[a, b]$,

then

$F:[a, b]\rightarrow R$ is differentiable on [a, b]

where $F(x)=\int_a^xf(t)dt$

Answer 1

Consider

$$f(x) = \begin{cases} -1, & -1\le x < 0 \\ 0, & x = 0 \\ 1, & 0 < x \le 1\end{cases}.$$

What is its antiderivative?