Product of two 2D smooth functions

[Telemachus]

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 $g(x,y)=g(y)$ for all x. In this situation is also ensured the differentiability of $h(x,y)$ by the differentiability of $f(x)$ and $g(y)$?

 

[andrewkirk]

Yes.

I suggest avoiding the word 'smooth' in this case though. It is generally reserved for functions that are infinitely differentiable. It happens to also be true that the product of two smooth functions is smooth.

By the way, in general a function $h:\mathbb R^2\to\mathbb R$that is partial differentiable with respect to both its arguments is not necessarily differentiable. But the function you have given above is in a particularly well-behaved subset of those functions, so it is differentiable.

 

[Telemachus]

Great, thank you! and yes, I am particularly interested in the situation where both functions are differentiable. It is somewhat restrictive, but is the case I am concerned with. I remember from my multivariate analysis curse (many many years ago) that differentiation in multivariable calculus becomes somewhat subtle. So I wasn't sure about it, and I was working on some numerical algorithms where this fact is important.