Mistake in (french) wikipedia on definition of limit?

Question

In the french wikipedia they say that (for $f:U\to \mathbb R$ where $U$ open) that $\displaystyle \lim_{x\to a}f(x)= \ell$ (where $a\in U$) if $$\forall \varepsilon >0, \exists \delta >0: \forall x\in U, |x-a|<\delta \implies |f(x)-L|<\varepsilon .$$ Isn't it wrong ? For example, take $\displaystyle f(x)=\boldsymbol 1_{\{0\}}(x)$. Then, $\lim_{x\to 0}f(x)=0$, but according to the french wikipedia definition, it doesn't converges to $0$ since if $\varepsilon <1$, if $\delta >0$, then for $x=0$, we have that $|x-0|<\delta $, but $|f(x)-0|=1>\varepsilon $. Is this a mistakes or is it a more general definition that is more restrictive ?

Answer 1

No, it is not wrong. Some authors use that definition of limit, with respect to which the value of $f$ at $a$ matters. You can recover the usual definition of limit from this one using the expression $\displaystyle\lim_{x\to a,\ x\neq a}f(x)$.

Of course, under this definition, whenever $a$ belongs to the domain of $a$, the limit $\lim_{x\to a}f(x)$ either doesn't exist or it is equal to $f(a)$.