Is the definite integral over a continuous function always finite? 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?

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