Is there a Heine cretierion of liminf of a function?

Question

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$.

Could you give me some intuition to the ugly definition? Is there a criterion using sequences in order to understand the $\liminf$?

Thanks

Answer 1

well, if you want sequences then it could go something like "there exists a sequence $x_n \rightarrow x_0$ such that $f(x_n) \rightarrow \liminf_{x_0} f$ and for any other sequence $x_n \rightarrow x_0$ with $f(x_n) \rightarrow g$ we have $g \geq \liminf_{x_0} f$

edit: by the way - if you're asking about an alternative definition it would be useful if you included the one that you know just so that I know I'm not rewriting the definition you struggle with