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How to link two curves?

I've come across the following problem a lot in analysis and physics, and I'm wondering what tricks people use to deal with it.

Let's say we have two smooth functions $f_1$ and $f_2$ defined on disjoint subsets of $\mathbb{R}$, say $(-\infty,a]$ and $[b,\infty)$ where $a

Usually $f_1$ and $f_2$ are something simple, like linear functions, and I could explicitly write it down... but I'm looking for a slicker (and quicker) argument.

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    Have you seen this question? http://math.stackexchange.com/questions/72608/how-to-link-two-curves2011-10-15
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    I disagree that this is a duplicate of the other question. One has to tweak Davide's argument from the other thread quite a bit in order to achieve what is asked here. Therefore I voted to reopen this question.2011-10-15
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    Ah, thanks. Personally, I'm satisfied with the answer in the other thread.2011-10-15

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