Function which the limit is the same for every point, but the function is not the same for every point

Question

Let $L \in \mathbb R$.
Let $f:\mathbb R\to\mathbb R$.
$\lim_{x\to x0}f(x)=L$ for every $x0\in \mathbb R$, so there exists $x\in \mathbb R$ such that $f(x) \neq L$.

I tried finding such function that for every point the limit is the same, but I couldn't find any function besides constant functions and then the second part of the claim is false.
Maybe this claim is false?

Answer 1

You can take$$f(x)=\begin{cases}0&\text{ if }x\ne0\\1&\text{ otherwise.}\end{cases}$$