My research is dealing with a functional $f(x,y): \mathbb{R}^n \times \mathbb{R}^k \to \mathbb{R}^+$, that I know is lower-semicontinuous in both $x$ and $y$. I would like the following to be true, but am having trouble proving it:
Conjecture: If $f(x_0,y_0) > M$, then there is some open neighborhood of $(x_0,y_0)$, call it $U$, such that $f(x,y) > M$ for all $(x,y) \in U$.
Is this true? Any help would be much appreciated.
edit: The particular functional is related to geodesic flow. See my comment below for more details.