Open condition given by inequality on functions

Question

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$.

The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ for which $f(x,y)<g(x,y)$ form an open subset of $D$?

I fear it's a trivial question, but I have a brainfreeze right now..

Thank you

Answer 1

Yes, these points are described by the open set $h^{-1}(0, \infty)$, where $h(x,y)= g(x,y)-f(x,y)$, and $h$ is smooth ( though continuous would be enough).