1)Let $f: [0,1] \to R$ be a continuous function such that $\mid f(x)-f(y) \mid \le \mid \sqrt{x}-\sqrt{y} \mid $ for all $ x,y \in [0,1]$. Then is $f$ absolutely continuous on $[0,1]$?
2) what about if $\mid f(x)-f(y) \mid \le \mid x^{1/3}-y^{1/3} \mid $ for all $ x,y \in [0,1]$.
and are both can be bounded variation? justify
These are current problems I am struggling. for me this. I think these function are uniformly continuous. but how can I use this fact to my problems?
please,help me out.