Definition of entropy of probability density function $f\ge 0$ is$$Ent(f)=\int_{\mathbb{R}} f(x)\log f(x)dx$$
My question is how this definition defines an integral on a set such that f is zero.$0\log0$ is $0$?or $-\infty$?
Also, is there a probability density $f$ such that $Ent(f)=+\infty$?