Constructing a function which fails the Mean Value Theorem

Question

I want to find an example of a function $f$ which is Riemann integrable on $[0,1]$ but does not satisfy the Mean Value Theorem. I need to explicitly show that $f$ does not satisfy the theorem.

Would the $\int_a^b f(x)\,d dx = F(b) - F(a)$ be a possible function?

Answer 1

Take $f(x)=1_{x\geq\frac{1}{2}}$. $f$ is monotone, so riemann integrable.