Continuity of a function

Asked Nov 14 '14 at 02:23

Active Nov 14 '14 at 14:07

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I've been trying to work this problem out I feel like I can't even figure out where to start. I was hoping someone could walk me through this.

Justify: The definition of continuity of any function $f: D\to R$ at a point a point $a \in D$ is equivalent to (i.e., true if and only if) the following statement holding:

The preimage under $f$ of any open interval containing $f(a)$ (i.e. $f^{-1} ((g,h)),$ where $g<f(a) <h$) contains the intersection of $D$ with some open interval $(c, d)$ containing $a$ (i.e., $(c,d)$ intersect $D$ is a subset of $f^{-1}((g,h)),$ where $c<a<d$).

edited Nov 14 '14 at 14:07

asked Nov 14 '14 at 02:23

MathManiac

Is $D$ a subset of the real numbers? Given that you've tagged it real analysis, I'd say it appears so. – Clayton Nov 14 '14 at 02:27
Yes, $D$ is a subset of the reals. – MathManiac Nov 14 '14 at 02:30
I think, this would help http://math.stackexchange.com/questions/61941/continuous-function-and-open-set – MA94554 Nov 14 '14 at 02:32
Is there anyway you could explain it to me by step? – MathManiac Nov 14 '14 at 02:37

Continuity of a function

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