Fundamental Theorem of Calculus, Part I

Fundamental Theorem of Calculus (Part I): Let $f:[a,b]\to\mathbb{R}$ be Riemann integrable and define $F(x)=\int_a^x f(t)\,dt \qquad (x\in[a,b]).$ Then $F$ is continuous on $[a,b]$. Moreover, if $f$ is continuous at a point $x_0\in(a,b)$, then $F$ is differentiable at $x_0$ and $F'(x_0)=f(x_0).$

This theorem is the precise link “differentiation undoes integration” (at points where $f$ is continuous). It also provides a systematic way to construct antiderivatives.