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The following are equivalent.

(i) f is continuous at c.

(ii) For every real ε > 0 there is a real δ > 0 depending on ε such that:


|x - c| < δ, |f(x) - f(c)| < ε.


Both (i) and (ii) are equivalent to

limx→c f(x) = f(c).

Intuitively, this corollary says that f is continuous at c if and only if f(x) is close to f(c) whenever x is close to c.

Last Update: 2006-11-05