Fundamental theorem of calculus/Riemann/Fact/Proof

Proof

Let ${}x$ be fixed. The difference quotient is

${}{\frac {F(x+h)-F(x)}{h}}={\frac {1}{h}}{\left(\int _{a}^{x+h}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right)}={\frac {1}{h}}\int _{x}^{x+h}f(t)\,dt\,.$

We have to show that for ${}h\rightarrow 0$, the limit exists and equals ${}f(x)$. Because of the Mean value theorem for definite integrals, for every ${}h$, there exists a ${}c_{h}\in [x,x+h]$ with

${}f(c_{h})\cdot h=\int _{x}^{x+h}f(t)dt\,,$

and therefore

${}f(c_{h})={\frac {\int _{x}^{x+h}f(t)dt}{h}}\,.$

For ${}h\rightarrow 0$, ${}c_{h}$ converges to ${}x$, and because of the continuity of ${}f$, also ${}f(c_{h})$ converges to ${}f(x)$.

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