Why do we use this class of functions? This is mainly concerned with studies coming from Gilbarg and Trudinger.  Elliptic Partial differential equations are of the form $Lu =f$, where $L = a^{ij}D_{ij} + b^iD_i + cu$, and most of the results come assuming that $f \in C^{\alpha}$, that is, $f$ is holder continuous, $0<\alpha<1$.  What the book doesn't really explain is why we consider holder continuous functions, and not some other class of functions.. for example $C^0, C^1$, etc.  I get that the estimates work for $C^{\alpha}$, but "because it works" isn't really a good explanation.
 A: The Holder space $C^{\alpha}(X)$ is stronger than $C(X)$ but weaker than $C^1(X)$. Overall, it is simply natural in the analysis of function spaces to consider such spaces when you're trying to figure out how "nice" a function is. The Holder seminorm for $0 < \alpha < 1$
$$ \|f\|_{C^{\alpha}} = \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}}$$
measures how much the function $f$ behaves somewhat like how $x^{\alpha}$ behaves near $x = 0$. Certainly a function like $x^{1/2}$ can be considered somewhat "nice" near $x = 0$ even though it fails to have a derivative at that point.
From a PDE standpoint, Morrey's inequality tells us that Sobolev spaces can be embedded into a certain Holder space, which again tells us how "nice" Sobolev spaces can end up being. In the context of your book, Gilbarg and Trudinger, proving regularity results for $C^{\alpha}$ is "sharper" than proving regularity in other more traditional spaces like $C^k$ with $k \in \mathbb{Z}^{+}$, because it tells us that we don't actually need functions to be that nice. 
