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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 $a<b$. I want to use a smooth function that is defined on all of $\mathbb{R}$ and agrees with $f_1$ on the first subset, and with $f_2$ on the second subset. How do I argue that such a function exists?

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.

Tony
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    Have you seen this question? http://math.stackexchange.com/questions/72608/how-to-link-two-curves – SBF Oct 15 '11 at 19:11
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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. – t.b. Oct 15 '11 at 19:57
  • Ah, thanks. Personally, I'm satisfied with the answer in the other thread. – Tony Oct 15 '11 at 23:04

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