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- #1

- Apr 13, 2013

- 3,841

I am looking at the theorem:

"$f:[a,b] \to \mathbb{R}$ integrable

We suppose the function $F:[a,b] \to \mathbb{R}$ with $F(x)=\int_a^x f$.If $x_0$ a point where $f$ is continuous $\Rightarrow F$ is integrable at $ x_0$ and $F'(x_0)=f(x_0)$".

There is a remark that the theorem stands only if $f$ is continuous and there is the following example:

$f(x)=\left\{\begin{matrix}

-1,-1 \leq x \leq 0\\

1,0 < x \leq 1

\end{matrix}\right.$

which is not continuous at $0$.

Then,according to the textbook, the function $F(x)$ is equal to $|x|-1$..But,why is it like that??And also why is $F$ not differentiable at $0$??