To my understanding, if $f$ is injective and $X$ is a continuous random variable, then $H(X | Y)$ with $Y = f(X)$ is not well-defined since the conditional probability distribution $p(x|y)$ on $X|Y=y$ is either not well-defined or delta function.
But I assume there is a more general framework that resolve this issue. How one can properly define the differential entropy for injective function (and functions that are injective on some particular regions)?