# Doubt regarding the function of complete statistic

If a statistic is complete, then any function of that statistic is complete. Is it true in general? For example, I know that for a gamma distribution $G(2,\theta)$, Sample mean will be a complete statistic. Then, can I say that any function of sample mean will be a complete statistic?

If $T$ is complete and $V=f(T)$, then, if $$\mathbb{E}_\theta[\Psi(V)]=c$$ for all $\theta$'s, this implies that $$\mathbb{E}_\theta[\Psi(f(T))]=\mathbb{E}_\theta[(\Psi\circ f)(T)]=c$$ for all $\theta$'s. Therefore, $\Psi\circ f$ is constant on the support of $T$ (i.e., the set of the possible values taken by $T$), $$(\Psi\circ f)(t)=\Psi(f(t))=c\qquad t\in\text{supp}\,T(\mathcal{X})$$hence$$\Psi(v)=c\qquad v\in\text{supp}\,fT(\mathcal{X}))$$which shows $V$ is complete.