Proving continuity and differentiability of a function at a point

Question

Let f be a real valued function satisfying

|f (x) − f (a)| ≤ C |x − a|ⁿ for some n> 0 and C > 0

(a) If n = 1, show that f is continuous at a

(b) If n > 1, show that f is differentiable at a.

Answer 1

Let $\varepsilon>0$ s.t. $n=1+\varepsilon$. Then, $$\frac{|f(x)-f(a)|}{|x-a|}\leq C|x-a|^\varepsilon.$$

Great ! $a$ is even a stationary point !