epsilon/delta definition alternative? The phrase "any epsilon greater than zero" has always seemed somewhat vague.
Question: is this an equivalent definition?
$\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. }0<|x−x_0|<δ \implies|f(x)−f(x_0)|<1/N$.

Another possible alternative to the epsilon, delta definition of continuity is Cauchy's original definition. Namely, $f$ is continuous at $x$ if for every infinitesimal $\alpha$ the difference $f(x+\alpha)-f(x)$ is similarly infinitesimal. See Cours d'Analyse.
