Assume that $f:[0,1]\to [0,1]$ is holder continuous with the constant $1/3$ and assume that for every $s\in [0,1]$ we have $$\limsup_{x\to s}\frac{|f(x)-f(s)|}{|x-s|^{1/2}}<\infty.$$  Does this implies that $$\sup_{s\neq t}\frac{|f(x)-f(s)|}{|x-s|^{1/2}}<\infty.$$