Differentiablity of certain composite function

Question

Let $I_1$ and $I_2$ be two closed bounded intervals. Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$.

Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ which is a diffeomorphism restricted to $B$, where $B$ is an open bounded box such that $I_1 \times I_2 \subseteq B$.

I want to define $\widetilde{W} (u,v) = W \circ \Phi^{-1} (u,v)$ which is well defined on $\Phi(B)$. I would like to extend it to be a function on all of $\mathbb{R}^2$ by defining it to be $0$ outside $\Phi(B)$.

My question is, from this does it follow that $\widetilde{W} (u,v)$ differnetiable everywhere? or do I need extra assumptions? (I guess my concern is what happens near the boundaries, does everything work out nicely so that this is the case?) Thank you very much.

Answer 1

The function $\widetilde W$ is a smooth function on the open set $\Phi(B)$ and is supported in the compact set$\Phi(I_1\times I_2)$. As a result it can be extended by 0 as you wish and this is simply due to the continuous canonical injection $$ C_{comp}^\infty(\Omega)\subset C_{comp}^\infty(\mathbb R^n), $$ for $\Omega$ open subset of $\mathbb R^n$.