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 $$\sup_{s\neq t}\frac{|f(x)-f(s)|}{|x-s|^{1/2}}<\infty.$$

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