When a random variable $X$ has only one possible outcome $x_0$, the probability density function at $X=x_0$ is infinite, while the probability density at other locations is zero. Then the p.d.f is exactly a delta function $\Pr(X=x) = \delta(x=x_0)$.
However, when I tried to calculate the entropy of the random variable, the problem arises. How can I calculate the integral $\int_{-\infty}^{+\infty}{\delta(x-x_0)\log\delta(x-x_0) \, dx}$?