Hi there. It is obvious that if you have two differentiable functions ##f(x)## and ##g(x)##, then the product ##h(x)=f(x)g(x)## is also smooth, from the chain rule.
But if now these functions are multivariate, and I have that ##h(x,y)=f(x)g(y)##, that is ##f(x,y)=f(x)## for all y, and similarly...