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let $E$ subset of $\mathbb R$ be compact and the function $f$ is continuous on $E$ and the function $g$ is integrable on $E$. prove that the function $fg$ is integrable.

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1. You could start accepting some answers. See here how it is done 2. Please read this thread on how to ask homework questions. 3. Note that $f$ is bounded. –  t.b. Oct 31 '11 at 14:43
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1 Answer

A continuous function $f$ on a compact set $E$ of $\mathbf R$ is bounded; that is, the sup-norm of $f$ is finite (i.e., $\|f\|_\infty < \infty$).

Now note that

$$\int |fg| \leqslant \|f\|_\infty \int |g| < \infty.$$

Also, follow t.b.'s advice please.

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