# $f(b)\leq f(a)+\omega(|b-a|)^\alpha$ with $\omega(0)=0$ and $\omega$ being differentiable implies that $f$ is smooth

Suppose $f,\omega:{\bf R}\to{\bf R}$ are functions with $\omega(0)=0$. Suppose for some $\alpha>1$, we have $$f(b)\leq f(a)+\omega(|b-a|)^\alpha\quad\hbox{for all } a,b\in{\bf R}\tag{*}$$ If $\omega$ is differentiable at $x=0$, show that $f\in C^\infty({\bf R})$.

The original problem is given as follows:

I think $\omega(|b-a|)^\alpha$ should be understood as $[\omega(|b-a|)]^\alpha$.

The condition (*) can be written as $$\frac{|f(x+h)-f(x)|}{h}\leq \frac{\omega(|h|)^\alpha-\omega(0)^\alpha}{h}.$$ This seems to imply the differentiability of $f$. But how would one expect that $f$ could be smooth?

$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
Since $\alpha>1$, the map $x\mapsto x^\alpha$ is differentiable at $x=0$. The inequality $$\frac{|f(x+h)-f(x)|}{h}\leq \frac{\omega(|h|)^\alpha-\omega(0)^\alpha}{h}.$$ implies that for any $x\in{\bf R}$, $$|f'(x)|\leq \omega(0)^{\alpha-1}\cdot\omega'(0)=0$$ which implies that $f$ is a constant.