Can a Riemann integrable function $f$ on $[-1,1]$ such that $F(x)=\int_{-1}^{x}f$ be differentiable at every point except for $x=0$?

Question

Is it possible to have a Riemann integrable function $f$ on $[-1,1]$ such that $F(x)=\int_{-1}^{x}f$ is differentiable at every point except for $x=0$?

My initial thoughts are that such function is not possible, but I can't find a way to demonstrate it.

Answer 1

With $$F(x) = \begin{cases} -1 & x < 0 \\ 1 & x \geq 0 \end{cases},$$ we have $$\int_{-1}^x F(t)dt = |x| - 1 $$ for all $x \in [-1,1]$, which gives such an example.