Consider two continuous functions $f_1:\mathbb R^n\to\mathbb R$ and $f_2:\mathbb R^n\to\mathbb R$. Define the set: $$ \mathcal S\triangleq \{x\in\mathbb R^n : f_1(x)<f_2(x)\}, $$ where the inequality is strict. Suppose that there exists at least one point in $\mathcal S$, i.e. the set is non-empty. I feel like it is then true that:
For any $x\in\mathcal S$, there exists a radius $r>0$ such that $\{y\in\mathbb R^n:\|y-x\|_2\le r\}\subseteq\mathcal S$ (i.e. there is a ball of some radius $r$ that fits into $\mathcal S$ at any of its points).
Does anyone know of a formal statement of this conjecture?
This could be stated as follows:
Let $X,Y$ be topological spaces, $f:X\rightarrow Y$ continuous and $U\subset Y$ open. Then $f^{-1}(U)\subset X$ is open.