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.

Proof sketch: Continuity of $F$ follows from boundedness of $f$ and the estimate $|F(x)-F(y)|=\left|\int_y^x f(t)\,dt\right|\le \|f\|_\infty |x-y|.$ For differentiability at $x_0$, compute the difference quotient : $\frac{F(x_0+h)-F(x_0)}{h}=\frac{1}{h}\int_{x_0}^{x_0+h} f(t)\,dt.$ If $f$ is continuous at $x_0$, then on small intervals $f(t)$ is close to $f(x_0)$, and a squeeze argument shows the quotient tends to $f(x_0)$.