I was thinking about the epsilon-delta definition of a limit and began to wonder if it would work with closed intervals instead of open intervals. That is, for some function $f$, would the folowing definition work? $$\forall \epsilon>0\,\exists\delta>0\,(0<\lvert x-c\rvert≤\delta \, \implies \lvert f(x)-L\rvert≤\epsilon) $$ I've been trying to think of a counterexample, but can't seem to find one.
edit: If it does work, why choose open intervals rather than closed?