I am trying to teach myself some calculus and statistics and I came across the idea of a probability density function which, as I understand, basically describes the probability per unit value of the random variable.
So it is analogous to densities we encounter in Physics and tells how much probability is concentrated near $x$ and to get the probability we have:
$$P(x) = f(x)\,\mathrm{d}x$$
where $f(x)$ is the density function and $\mathrm{d}x$ is the differential.
Now, from laws of probability we have:
$$ \int _{-\infty}^{\infty} f(x)\,\mathrm{d}x = 1 $$
The thing that I cannot get my head around is that this PDF can take on value of infinity for certain values of $x$ and still have a finite integral of $1$. How is that possible?