Is uniform boundedness is related to Hölder continuity of a function?
I mean is it necessary to prove first uniform boundeness to prove the Hölder continuity of a function?
Also tell me the practical meaning of cone of influence. I know it is already being explained on this site, but I wants to learn the practical significance of it if possible.
A Hölder continuous function may be unbounded. For example, for any $\alpha\in (0,1]$ the function $f(x)=|x|^{\alpha}$ is unbounded on $\mathbb R$; yet it is Hölder continuous with exponent $\alpha$.
However, if the domain of a Hölder continuous function is a bounded subset of $\mathbb R$ (or of $\mathbb R^n$), then the function is also bounded. This is true for all uniformly continuous functions, actually. Here is a proof by contradiction: suppose there is a sequence $x_n$ such that $|f(x_n)|\to \infty$. Since $x_n$ is bounded, it has a convergent subsequence $x_{n_k}$. This subsequence is Cauchy. Use uniform continuity to conclude that $f(x_{n_k})$ is also Cauchy. But every Cauchy sequence is bounded, a contradiction.