Solutions

Notice that

$$\left|\frac{f(x)-f(0)}{x}\right|=\left|\frac{f(x)}{x}\right|\ge |x|^{\beta-1}$$

and

$$\lim_{x\to 0}|x|^{\beta-1}=\infty$$

since $0<\beta<1$. Hence,

$$\lim_{x\to 0}\frac{f(x)}{x}$$

does not exist, and $f$ is therefore not differentiable at $0$.