Counterexample of the almost-inverse of the Fundamnetal Theorem of Calculus(Lebesgue). Can anyone give me a counterexample to the following statement:

Suppose $F \colon [0,1] \to \mathbb{R}$ is continuous and differentiable almost everywhere, then $F(b)-F(a)=\int_a^b F'(t)\, \text{d}t$. 

I guess let the discontinuity be $\mathbb{Q} \cap[0,1]$ may help.

The Cantor function is a counter-example. It's continuous and has derivative $0$ almost everywhere, yet $F(1)-F(0)=1-0=1 \neq 0=\int_0^10 \, dt=\int_0^1F'(t) \, dt$.
