While doing the following practice question, I got stuck at the proof of uniform continuity of this function. (I know it should be uniformly continuous iff $0< \alpha < 1$)
We can easily show that it will be uniformly continuous on $[0,1]$, and I believe that if it is also uniformly continuous on $(1,+\infty)$, then we are done. But how should we prove that? Also, how can we show that if $\alpha>1$ or $\alpha<0$, the function is not uniformly continuous?