I am working on the following exercise:
Let $X$ be a discrete RV with values in $\mathbb{R}$ and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an arbitrary measurable function.
(a) Show that, in general, $H(f(X)) \le H(X)$.
(b) When do we have equality $H(f(X)) = H(X)$?
(c) When do we have $H(f(X)) = 0$?
I have done the points a) and b), but I do not understand what I am supposed to do in c). Could you explain?
Instead of giving a complete answer, let me illustrate with an example. Suppose the random variable $X$ takes two values $a$ and $b$ with probabilities $p$ and $q$. Then $$ H(X) = - p\log p - q \log q .$$ Now if $f(a) \ne f(b)$ then $H(f(X)) = H(X)$. But if $f(a) = f(b)$ then $f(X)$ becomes a random variable taking only one value, and $$ H(f(X)) = - (p+q)\log (p+q) .$$ In this particular case, $H(f(X)) = 0$ because $p+q = 1$. But I wrote it this way so that you can see how to generalize.