Define a function $f_{n}$ on $\mathbb{R}$ as $$ f_{n}(x)=\left\{\begin{array}{ll} 0 & \text { if } x=0 \\ x^{n} \sin \frac{1}{x} & \text { elsewhere } \end{array}\right. $$ where $n$ is a positive integer.
(a) Prove that $f_{n}$ is continuous on $\mathbb{R}$ for all $n$.
(b) Is $f_{n}$ is differentiable at $x=0$ ? If so, find $f_{n}^{\prime}(0)$.
The second part seems too obvious after applying the epsilon-delta rule using the upper bound $1$ for $sin(\frac{1}{x})$ function $\rightarrow$ $f_{n}^{\prime}(0) = 0$; however, the same technique did not work for the part $a$ in which the desired epsilon-delta inequalities do not match precisely. Is there any alternative method for solving the first part?