Counterexample of the almost-inverse of the Fundamnetal Theorem of Calculus(Lebesgue).

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