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What does $C^\infty([a,b]; \mathbb{R})$ denote? I know it's a set of functions $[a,b] \to \mathbb{R}$. I think the $C$ stands for continuous. What does the $^\infty$ mean here?

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    I$n$fi$n$itely differe$n$tiable-- smooth.2012-11-11

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Smooth (ie, infinitely differentiable) functions from $[a,b]$ into $\mathbb{R}$.

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    Existence of one sided derivatives is sufficient here. In general, I suppose the domain should be such that it is 'sufficiently rich' around the boundary to uniquely define$a$derivative.2012-11-11
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There are two natural definitions.

  1. It could mean continuous functions on $[a,b]$ which are smooth in the interior $(a,b)$ and so that all derivatives have limits at the end point $a$ and $b$.
  2. It could also mean functions on $[a,b]$ that can be extended to infinitely differentiable functions on a larger, open interval $(a-\varepsilon,b+\varepsilon)$.

Fortunately, the two are equivalent. It is rather obvious that a function which is smooth in the second sense is smooth in the first sense. The converse can be proved by appeal to the theorem that, for every sequence $(c_n)$ of real numbers, there is an infinitely differentiable function $g$ defined on a neighbourhood of $a$ so that $g^{(n)}(a)=c_n$ for all $n$. Given a function $f\in C^\infty([a,b])$ in the first sense, put $c_n=\lim_{x\to a}f^{(n)}(x)$, pick a $g$ by the theorem mentioned, and extend $f$ by using $g$ instead to the left of $a$. Do the same at $b$. The result is a smooth function on a bigger interval, so $f$ smooth on $[a,b]$ in the second sense.

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    Thanks! I just found it, it is called Borel's lemma. However,the construction seems a little artificial to me, in that it just defines a function $g$ whose derivatives match the limits of $f$'s derivatives. But since $f$ and $g$ need not match on the overlapping domains, one could have obtained the same result just by defining the derivative at the limit as the limit of the derivative and showing that it has the appropriate Frechet-like property there?2012-11-11