I saw this problem in Charles Chapman Pugh's Mathematical Analysis :
Consider a modulus of continuity $\mu (s) = H^\alpha$ where $0 < \alpha\leq1$ and $0 < H < \infty$. A function with this modulus of continuity is said to be $\alpha$-Holder, with $\alpha$-Holder constant $H$ .
$(a)$ Prove that the set $C^\alpha(H)$ of all continuous functions defined on $[a , b]$ which are $\alpha$-Holder and have $\alpha$-Holder constant $\leq$ H is equicontinuous .
$(b)$ Replace $[a , b]$ with $(a , b)$ . Is the same thing true?
$(c)$ Replace $[a , b]$ with $\mathbb{R}$ Is it true?
$(d)$ What about $\mathbb{Q}$?
$(e)$ What about $\mathbb{N}$ ?
But basically here's my problem :
Part $(a)$ is easy as you can control $|f(x)-f(y)|\leq H|x-y|^\alpha$ everywhere by controlling $|x-y|$. But I don't know the story will be different for next parts or not ?! Did we really used any feature of closed interval $[a,b]$ in part $(a)$ ?
No, you did not.
It will not be any different. The same reasoning you applied for $[a,b]$ works for every metric space.