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The phrase "any epsilon greater than zero" has always seemed somewhat vague.

Question: is this an equivalent definition?

$\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. }0<|x−x_0|<δ \implies|f(x)−f(x_0)|<1/N$.

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  • $\begingroup$ Yes this is equivalent due to the Archimedean property of the natural numbers. I think we just like our $\varepsilon$. $\endgroup$ – Cameron Williams Feb 8 '14 at 2:23
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    $\begingroup$ In fact, you can just use every second natural number if you want to be more economical... $\endgroup$ – copper.hat Feb 8 '14 at 2:27
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    $\begingroup$ And to avoid Greek letters altogether, we can say that for any positive integer $N$ there is a positive integer $Q$ such that whenever whenever $|x-x_0|\lt \frac{1}{Q}$, $\dots$. $\endgroup$ – André Nicolas Feb 8 '14 at 2:33
  • $\begingroup$ What does second natural number mean? $\endgroup$ – Hawk Feb 8 '14 at 4:38
  • $\begingroup$ He means you can use only even natural numbers: "$\forall$ even natural numbers $N>0$ …" $\endgroup$ – Lee Mosher May 2 '14 at 13:07
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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.

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