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What is the importance of defining the classes $C^k$ of functions ? Wouldn't be more natural defining the classes of functions differentiable k-times, with k=0,1,2,... Why the continuity of the last derivative is so important to justify a specific definition ?

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Somes theorems in analysis genuinely require continuity of the derivative (e.g. you may have problems with the Fundamental Theorem of Calculus) –  Yemon Choi Dec 3 '12 at 5:08
See also this MO post: mathoverflow.net/questions/66462/… –  Todd Trimble Dec 3 '12 at 6:10
You get a well behaved Banach space only if you get one when $k=0$. –  Deane Yang Dec 3 '12 at 12:29
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1 Answer

Among the properties that make $C^k$ a nice category, and that have no analog for just differentiable maps, I'd list these (some of whom have already mentioned in the comments):

1. The total differential theorem. To check that a function of several variables is $C^k$ becomes a matter of partial differentiability in each single variable (and continuity of the partial differentials in all of them).

2. The local inverse theorem. The core of local differential calculus: checking that a local property holds locally at a point just requires verifying that the analogous property holds for the differentials at that point in the corresponding linear category. This holds for being a local diffeo, a local immersion or submersion; for transversality of mappings, &c.

3. Characterization. (Converse of Taylor formula). A map is $C^k$ if and only if it has a polynomial expansion of order $k$ "locally uniformly", with continuous coefficients.

4. Whitney extension theorem. With a natural notion of $C^k$ smoothness on closed sets, every smooth map on a closed set extends on the whole space.

5. Ordinary Differential Equations. $C^k(I)$ is a convenient space for well-posed ODE, starting with the problem of finding an antiderivative, given by the Fundamental Theorem of Calculus. Compare with the difficulty of characterising derivatives of everywhere differentaible functions (see this MO question). Note however that $C^k$ is not so a convenient setting for PDE's, e.g. in potential theory.

6. Functional Analysis. For any open set $\Omega\subset\mathbb{R}^n$ the class $C^k(\Omega)$ is a Fréchet space, embedded as a closed subspace into $C^0(\Omega,\mathbb{R}^N)$ taking a function into its $k$-jet. Something similar also works for the class of $C^k$ maps between $C^k$ manifolds, that can be embedded in the space of $C^0$ maps in the $k$-jets.

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I would also add Clairaut's theorem to the list: $\frac{\partial^2 F}{\partial x_i \partial x_j} = \frac{\partial^2 F}{\partial x_j \partial x_i}$ automatically true for $C^2$ functions, but can fail for functions that are merely twice differentiable. (Related to this is the fact that for a $C^1$ function, the strong derivative and the weak derivative agree, whereas this is not necessarily the case for functions that are merely everywhere differentiable.) –  Terry Tao Dec 3 '12 at 16:47
And to add to point 6: the function space $C^k$ offers not only a qualitative notion of regularity, but also a quantitative notion of magnitude (the global or local $C^k$ norms), and thence a notion of convergence. The notion of a sequence or series being convergent in the $C^k$ topology can be used, among other things, to demonstrate the k-times differentiability of a limit or infinite series (cf. the Weierstrass M-test). This can't be done if one restricts attention only to the notion of k times differentiability without any further control on the k-fold derivative. –  Terry Tao Dec 3 '12 at 16:50
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