Vtyjkyuk

I have trouble in the following problems:

1. 
   

   Let $f:(0,infty)tomathbbR$ be a function defined by $f(x)=x^alphasin(x^alpha)$, where $alphainmathbbR$.

   
   
   

   Find the range of $alpha$ so that $f$ is uniform continuous on $(0,infty)$.

   
   


2. 
   

   Let $g:(0,infty)tomathbbR$ be a function defined by $g(x)=x^betacos(x^beta)$, where $betainmathbbR$.

   
   
   

   Find the range of $beta$ so that $f$ is uniform continuous on $(0,infty)$.

   
   

Using the Continuosly Extension Theorem and Bounded Derivative property, i guess $0<alpha,betalefrac12$.

But, i can't prove the other case that $f$ is not uniformly continuous unless $0<alpha,betalefrac12$.

Give some hint or advice. Thank you.

Hint 1:

For $alpha > 1/2$, take $x_n = (npi + 1/n)^1/alpha$ and $y_n = (npi)^1/alpha$ and show that as $n to infty$ we have $|x_n-y_n| to 0$ but $|f(x_n) - f(y_n)| notto 0$.

Hint 2:

Using the binomial expansion

$$|x_n - y_n| = (npi)^1/alphaleft[left(1 + frac1n^2piright)^1/alpha - 1right]= (npi)^1/alphaleft[1 + frac1alpha n^2pi + O(n^-4)-1right]$$