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eg. is $$\int_a^b{f(x)}\mathrm{d}x$$ always finite for a continuous function $f:\mathbb R\rightarrow\mathbb R$ ?

If not are there any particular constraints that $f$ must obey for this to be true?

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Yes, since $f$ is bounded on $[a, b]$. Someone will give a more detailed answer I'm sure! – Dylan Moreland Dec 12 '11 at 22:22
up vote 6 down vote accepted

Since $[a,b] = \{ x \in \mathbb{R} \;|\; a \leq x \leq b \}$ is a compact set, any continuous function restricted to $[a,b]$ attains a minimum and maximum value on $[a,b]$ (this is the extreme value theorem). Thus there exists numbers $m,M \in \mathbb{R}$ such that $m \leq f(x) \leq M$ for all $a \leq x \leq b$.

Thus we have $$ m(b-a) = \int_a^b m\;dx \leq \int_a^b f(x)\;dx \leq \int_a^b M\;dx = M(b-a) $$

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