In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. Examples are a two-headed coin, a die that always comes up six. This doesn't sound very random, but it satisfies the definition of random variable.
The degenerate distribution is localized at a point x in the real line. On this page it is enough to think about the example localized at 0: that is, the unit measure located at 0.
The cumulative distribution function of the degenerate distribution is then the Heaviside step function:
Status of its PDF
As a discrete distribution, the degenerate distribution does not have a density.
P.A.M. Dirac's delta function can serve this purpose. But a serious theory awaited the invention of distributions by Laurent Schwartz.
NB: There is an unfortunate ambiguity in the meaning of the word distribution. The meaning given to it by Schwartz is not the meaning of the word distribution in probability theory.