Is there significant differences between Holder continuous function with exponent smaller and bigger than 1/2

Question

I've been reading about some materials regarding random series. And Holder continuity came up a lot and 1/2 seems like a magic number keep showing up in assumptions and theorems. Is there some significant differences between Holder continuous function with exponent smaller and bigger than 1/2. Such as functions will be constant with exponent bigger than 1 and differentiable if equal to 1.