I've been trying to understand this for a while now, but can't seem to get it. This question only refers to a discrete random variable X and some function $f$ in $\mathbb{R}$.
I've been told that $f(X)$ being an injective function is enough for the statement to hold.
But isn't $H(f(X)) = H(X)$ if the function is injective? This says nothing about the entropy being $0$. I would have assumed that it is only equal to $0$ if the function is injective AND the random variable already has $0$ entropy.
Am I completely off track here?
you're right, since if $f$ is injective, it can't change the probability density of the new random variable $Y = f(X)$. Are you sure the questions asks $f$ to be injective?
For a simple counterexample, let $f(x) = x$. Now, $H(f(X)) = H(X)$, an if $H(x) \neq 0$, then $H(f(X)) \neq 0$