density transformation preserving differentiability and continuity?

Question

Let X be a continuous r.v. with the density, $f_X(x)$, and let $Y=g(X)$. Suppose $f_X(\cdot)$ and $g(\cdot)$ are continuously differentiable. Could we prove that the density of Y, $f_Y(y)$, is continuously differentiable? It seems OK if g is one-to-one. How about $X$ is an n-dimensional random vector, $Y$ is an m-dimensional random vector , and $m<n$?