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- Apr 13, 2013

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Let $f:[a,b] \to \mathbb{R}$ bounded.We suppose that f is continuous at each point of $[a,b]$,except from $c$.Prove that $f$ is integrable.

We suppose that $c=a$.

$f$ is bounded,so $\exists M>0$ such that $|f(x)|\leq M \forall x$

Let $\epsilon'>0$.We pick now a $x_0 \in (a,b)$ such that $x_0-a< \epsilon'$.

Let $\epsilon''>0$

As $f$ is continuous at $[x_0,b]$ ,it is integrable.

So there is a partition $P$ of $[x_0,b]$ such that: $U(f,P)-L(f,P)<\epsilon''$

Now we consider the partition $P'=\{a\} \cup P$ of $[a,b]$.But,why do we take this partition??Aren't there any other points between $a$ and $x_0$ ??