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I just came across the Deuflhard & Bornemann text on Scientific Computing with ODEs where they write, for example:

$f \in C(\Omega, \mathbb{R}^d)$

In other places they use $C^1(\cdot,\cdot)$.

Easy question: what does the $C$ / $C^n$ refer to?

Thanks in advance.

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    Can somebody collect the content of all these comments into an answer? :)2011-12-05

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  • $C(X,Y)$ is the set of all continuous functions from $X$ to $Y$.
  • $C^k(X,Y)$ is the set of all $k$ times continuously differentiable functions from $X$ to $Y$.
  • $C^\infty(X,Y)$ is the set of all smooth functions from $X$ to $Y$.
  • $C^\omega(X,Y)$ is the set of all analytic functions from $X$ to $Y$.

If $Y$ is omitted it usually means that it's $\mathbb R$, but in some contexts it can also be $\mathbb R^n$, $\mathbb C$ or even $\mathbb C^n$.