Fundamental theorem of calculus/Riemann/Fact

Fundamental theorem of calculus

Let ${}I$ denote a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous function. Let ${}a\in I$, and let

${}F(x):=\int _{a}^{x}f(t)\,dt\,$

denote the corresponding integral function.

Then ${}F$ is

differentiable, and the identity

${}F'(x)=f(x)\,$

holds for all

${}x\in I$.

Proof, Write another proof