Characterization of entropy of probability distributions on the real line

A characterization of the entropy —∫ f log f dx of a random variable is provided. If X is a random variable and Y = T(X), where T is a piecewise monotonic function, the entropy difference H [X] — H [Y] is first characterized as the expectation of a function with reasonable properties. After entropy differences are characterized, zero entropy is assigned to the random variable which is uniformly distributed on the unit interval. In the course of the derivation, a characterization of Kullback's directed divergence is also obtained