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I just asked a specific homework question on this topic, but I want a more general explanation for how to go about proving continuity with this method.

I can't even wrap my head around what the proof is really saying, let alone figure out the steps to prove a specific function is continuous.

Edit: okay, here is a specific example. Maybe a walkthrough would be easier for you guys to help me out:

Prove that $g(x) = \frac{1}{x^2 - 4}$ is continuous on $\mathbb{R} \setminus \{ -2, 2\}$.

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    I'm not entirely sure what you're asking. If you want to understand how $\varepsilon$-$\delta$-proofs work, you could do worse than looking at *explicit* examples. You can find quite a few of these by searching on the site, e.g. [using this search](http://math.stackexchange.com/search?q=%22epsilon-delta%22).2011-12-05
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    And maybe [this is what you're looking for?](http://math.stackexchange.com/q/15963/)2011-12-05
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    You should study this answer, found in the search that t. b. suggests: http://math.stackexchange.com/a/11884/78502011-12-05
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    eps-delta is traditionally hard because it is one of the first times you encounter a statement with nested exists and for-all quantifiers. The problem is that unless you can specify a specific problem you are having I don~t think anyone here will come up with an explanation that is much better then the one in your textbook.2011-12-05
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    The $\epsilon$-$\delta$ is more unpleasant than usual in this case. In outline it is not hard, but there are petty details to worry about "near" $\pm 2$. Showing continuity at (say) $x=1.9$, or some other concrete number, would enable you to concentrate on the basic idea of the proof.2011-12-05

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