Integral from infinity to infinity My physics professor today wrote on the blackboard:
$$ \int_{\infty}^{\infty} f(x) dx = 0 $$
for every function $f$. 
And the proof he gave was:
$$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a} f(x) dx + \int_{a}^{\infty} f(x)dx = - \int_{a}^{\infty} f(x) dx + \int_{a}^{\infty}f(x)dx = 0$$
However I'm still not convinced, for me an integral from infinity to infinity has no meaning. Therefore, what I'm asking is: does the above equations make sense? If not, are there cases where they do make sense? I'm thinking about functions that converge to 0 in $+\infty$.
EDIT: Actually, the function f considered was a density, i.e.:
$$ \int_{-\infty}^{+\infty} f(x)dx = 1 $$
and $f(x) \geq 0$ for all $x$. 

This is not necessarily true. Take the following example;
$$\int_a^{2a}\frac1x\mathrm{d}x=[\ln{|x|}]_a^{2a}=\ln{(2)}$$
If we take $a\to\infty$ then the integral becomes
$$\int_\infty^\infty\frac1x\mathrm{d}x=\ln{(2)}$$ 
as the integral is constant for all $a\in\mathbb{R}$. What I guess your professor meant was that
$$\lim_{a\to\infty}\int_a^a f(x)\mathrm{d}x=0$$
which is trivially true as the LHS is constantly zero.
