multiplying by a $C^\infty$ function If $f \in C^\infty$ and $g$ is a real valued function can we say anything about their product? In particular is $fg \in C^\infty$ or maybe if we stipulate $g$ has compact support can we make the claim? What's a good way to look at this problem? Im looking to understand this as part of a derivation for a weak (FEM) formulation for Stokes Flow.

The only thing you can say that $fg$ has the same smoothness as $g$. That is, if $g$ is in $C^k$, then $fg$ is in $C^k$ as well.

If $f$ in addition has compact support, then $g\in C^k$ implies that $fg$ is in $C^k$ with uniformly continuous derivatives up to order $k$. Similarly, if $g$ is in some Sobolev space $W^{m,p}$, then $fg\in W^{m,p}$ as well.