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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?

BlizzardWalker
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    Induction would work. You need to first show that $f_1(x)=x\sin(1/x)$ is continuous on $\mathbb{R}$. Then, if, by induction hypothesis, you have $f_n(x)=x^n\sin(1/x)$ being continuous on $\mathbb{R}$, and you know that the product of continuous functions is continuous... – C Squared May 04 '21 at 03:53
  • @CSquared How can you show that $f_{n}(x) = x^n sin(\frac{1}{x})$ is a continuos function on any real number, using the epsilon-delta technique? I still can not figure out – BlizzardWalker May 04 '21 at 04:00
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    Oh never mind, found the problem: https://math.stackexchange.com/questions/274594/epsilon-delta-proof-that-fx-x-sin1-x-x-ne-0-is-continuous Thanks for pointing out the induction process, appreciate it! – BlizzardWalker May 04 '21 at 04:00

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