# Fundamental theorem of calculus/Riemann/Section

## Definition

Let ${\displaystyle {}I}$ denote a real interval, let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a Riemann-integrable function, and let ${\displaystyle {}a\in I}$. Then the function

${\displaystyle I\longrightarrow \mathbb {R} ,x\longmapsto \int _{a}^{x}f(t)\,dt,}$
is called the integral function for ${\displaystyle {}f}$ for the starting point ${\displaystyle {}a}$.

This function is also called the indefinite integral.

The following statement is called Fundamental theorem of calculus.

## Theorem

Let ${\displaystyle {}I}$ denote a real interval, and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a continuous function. Let ${\displaystyle {}a\in I}$, and let

${\displaystyle {}F(x):=\int _{a}^{x}f(t)\,dt\,}$

denote the corresponding integral function. Then ${\displaystyle {}F}$ is differentiable, and the identity

${\displaystyle {}F'(x)=f(x)\,}$

holds for all ${\displaystyle {}x\in I}$.

### Proof

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

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

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

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

and therefore

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

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

${\displaystyle \Box }$