What is the importance of defining the classes $C^k$ of functions ? 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 ?
 A: 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.
