Sufficient statistic for continous monotone function of parameter.

Question

Suppose we are estimating $\tau(\theta)=\theta e^{-\theta}$ from $X_1,...,X_n \sim G(\theta,r)$ (G is the gamma distribution) then it is easily shown that $T=\sum_{i=1}^n \ln(X_i)$ is sufficient for $\theta$ and $\exp(T)$ also. I'm wondering, is a continuous function of a sufficient statistic also sufficient and if $T$ is sufficient for $\theta$ is it also sufficient for $\tau(\theta)$?