The Fundamental Theorem of Real Analysis

The Fundamental Theorem of Real Analysis (Part I)

Let $f:D\to \mathbb{R}$ be a real function on a closed interval $D=[a;b]$.

If $f$ is Riemann-integrable, then $F:D\to \mathbb{R}$ defined by the definite integral

$$F(x)\stackrel{\text{def}}{=}{\int }_a^{x}f$$

is continuous. Furthermore, if $f$ is continuous at some $c\in D$, then $F$ is an antiderivative of $f$ at $c$, i.e. $F^{\prime }(c)=f(c)$.

PROOF

According to the definition of the derivative we need to prove

$$\underset{\Delta x\to 0}{\lim }\frac{F(x+\Delta x)-F(x)}{\Delta x}=f(x)$$

For all $c,x_0,\Delta x\in [a;b]$, where $x_0+\Delta x\in [a;b]$, it holds that

$$\begin{array}{rl}F(x_0+\Delta x)-F(x_0)& ={\int }_c^{x_0+\Delta x}f(t)\mathrm{dt}-{\int }_c^{x_0}f(t)\mathrm{dt}\\ & ={\int }_c^{x_0+\Delta x}f(t)\mathrm{dt}+{\int }_{x_0}^{c}f(t)\mathrm{dt}\\ & ={\int }_{x_0}^{c}f(t)\mathrm{dt}+{\int }_c^{x_0+\Delta x}f(t)\mathrm{dt}\\ & ={\int }_{x_0}^{x_0+\Delta x}f(t)\mathrm{dt}\end{array}$$

The mean value theorem for definite integrals says that there is at least one $\xi \in [x_0;x_0+\Delta x]$ such that

$${\int }_{x_0}^{x_0+\Delta x}f(t)\mathrm{dt}=f(\xi )(x_0+\Delta x-x_0)=f(\xi )\cdot \Delta x$$ $$\frac{F(x_0+\Delta x)-F(x_0)}{\Delta x}=f(\xi )$$

We now take the limit as $\Delta x\to 0$.

$$\underset{\Delta x\to 0}{\lim }\frac{F(x_0+\Delta x)-F(x_0)}{\Delta x}=\underset{\Delta x\to 0}{\lim }f(\xi )$$

The left-hand side is just the derivative of $F$ at $x_0$. Since $\xi$ is between $x_0$ and $x_0+\Delta x$, it must approach $x_0$ for $\Delta x\to 0$. This means that

$$\underset{\Delta x\to 0}{\lim }f(\xi )=f(x_0)$$

and so we have

$$\underset{\Delta x\to 0}{\lim }\frac{F(x_0+\Delta x)-F(x_0)}{\Delta x}=f(x_0)$$

We have thus proven

$$F^{\prime }(x_0)=f(x_0)$$

The Fundamental Theorem of Real Analysis (Part II)

If $f:[a;b]\to \mathbb{R}$ , then any antiderivative $F:[a;b]\to \mathbb{R}$ of $f$ can be used to calculate its definite integral as follows:

$${\int }_a^{b}f(x)\mathrm{dx}=F(b)-F(a)$$

NOTATION

The expression $F(b)-F(a)$ is usually shortened to just $F(x){|}_a^b$.

PROOF

TODO