Fundamental Theorem of Calculus, Part I in One Dimension

Let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$. Assume that $f$ is continuous at every point $x\in[a,b]$. Then for every $x\in[a,b]$, the Riemann integral $F(x)=\int_a^x f(t)\,dt$ is well defined. For every $x\in(a,b)$ the function $F$ is differentiable at $x$ and $F'(x)=f(x).$