Integrals

Let $f:[a,b] \to \R$ be Riemann integrable. Let $F:[a,b]\to\R$ be $F(x)= \int_{a}^{x}f(t)dt$.

Then $F$ is continuous, and at all $x$ such that $f$ is continuous at $x$, $F$ is differentiable at $x$ with $F'(x)=f(x)$.

$$F''(x)=f'(x)$$