Indefinite integral which is not differentiable

Question

I would like to know if there exists an indefinite integral which is not differentiable? Is this possible? That is, I want to know if there exists a real function F defined in a interval $[a,b]$ by the equation $$F(x)=F(a)+\int_a^x f(t) dt $$ where $f:[a,b]\rightarrow \mathbb{R}$ is only Riemann integrable, that $F$ is not differentiable of such a function? Could you give an example?

Answer 1

The fundamental theorem of calculus says that $F$ is continuous, and differentiable at every point where $f$ is continuous. So to get an example, you must take $f$ discontinuous. For example, if $f(x)=0$ for $x<0$ and $f(x)=1$ for $x\ge0$, then (with $a=0$) we get $F(x)=0$ for $x<0$ and $F(x)=x$ for $x\ge0$, so $F$ is not differentiable at the point $x=0$.