Hölder condition : Simple use case This may sound a newbie question,
I would like to hear a simple example of usefulness of Hölder condition i.e. how it helps in researching the function's properties.
 A: I am not sure what you are specifically looking for, but here are some scenarios where the the Hölder condition comes into play.
Suppose we have a function that satisfies the Hölder condition for $\alpha \geq 1$. Then we already satisfy a weaker condition called Lipschitz continuity. This condition is essential for two major theorems: the Picard-Lindelöf theorem and the Banach fixed point theorem. I would recommend looking at them.
There is a very important theorem in Real Analysis called the Arzela-Ascoli theorem. It says that if we have a family of functions $\{f_n: [a,b] \to \mathbb{R}\}_{n \in \mathbb{N}}$ such that the family is uniformly bounded and equicontinuous, then there is a subsequence that converges uniformly. Now, suppose our family of functions additionally satisfied the Lipschitz condition for a fixed constant. The Arzela-Ascoli theorem tells us that the limit function is Lipschitz and has the same constant. 
On a side note, it turns out that the set of Hölder continuous functions of order $\alpha$, denoted by $C^{\alpha}([a,b])$ is a Banach space under the norm $||f|| = |f(a)| + M_f$ where $$M_f = \sup_{s \neq t}{\frac{|f(s) - f(t)|}{|s-t|}}.$$ This was a question I had to prove in a homework a few months ago and it was pretty fun to solve.
