5.9 Proof of Fundamental Theorem of Calculus

The difference quotient of $F$ is

To prove that $F^{\prime}(x)=f(x),$ we need to prove that the right-hand side converges to $f(x)$ as $h\rightarrow 0$. Given that $f$ is continuous, $f(t)-f(x)\rightarrow 0$ as $h\rightarrow 0$ and ${{1}\over{h}}\int_{x}^{x+h}f(x)dt=f(x)$, so

hence $F^{\prime}(x)=f(x)$.