$f$ attains its minimum, delta definition

Question

I just want to check if the following "definition" makes sense and is correct:

A function $f: X \rightarrow \mathbb{R}$ attains its minimum if $\exists \delta > 0$ such that $f(x) \geq \delta$, for all $x \in X$.

Answer 1

Note: This is a formulation of the comments.

For any general mapping $f:X \to \mathbb{R}$, the restriction $\delta > 0$ must be removed. However, in your case of Lebesgue number lemma, the function $f$ is a finite-sum of metrics, which is never negative. The proof also shows that $f \neq 0$, so it is safe to assume that $\delta > 0$.

Furthermore, the definition you are using is the global minimum, which is different from local minimum when $X$ is endowed with some topology.