Let $X$ be a metric space and $f:X\to\mathbb{R}$ be lower semicontinuous (LSC) at $x\in X$, i.e. $\liminf_{y\to x} f(y)\geq f(x)$. A function is called LSC if it is pointwise LSC.
For my thesis I have to prove that if $f$ is LSC then $\liminf_{y\to x} f(y)=f(x)$ for all $x\in X$. So I assume by contradiction that there is an $x\in X$ such that $\liminf_{y\to x} f(y)>f(x)$. But why does it violate the LSC property? For example, the indicator $\mathbf{1}_{\mathbb{R}\setminus\{0\}}$ should not emit any problem when I use the usual definition $\liminf_{y\to x} f(y) :=\lim_{\varepsilon\downarrow 0} \inf_{y\in B_{\varepsilon}(x)\setminus\{x\}}f(y)$.
Any help is appreciated!