Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that $$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\alpha} \leq C \varliminf_{r \rightarrow 0^+}\frac{\sup_{x_1,x_2\in B(x,r)} |f(x_1)-f(x_2)|}{r^a} $$ for some positive constant $C$?