Proving a a function is continuous using ε and δ definition

Question

how would one go about proving the continuity of \begin{equation} f_n(x)=\begin{cases} \sin^2(\pi/x) & \text{ for } \frac1{n+1} \leq x\leq\frac1n\\ 0 & \text{otherwise} \end{cases} \end{equation}

I've tried using that $f_n(x)$ is continuous at c if and only if for every $ε>0 $, $∃ δ>0$ such that

$$|x-c|<\delta ⟹ |f(x)−f(c)|<\epsilon$$

so

$$|\sin^2(\pi/x) - \sin^2(\pi/c)| < \epsilon$$

For some $|x-c|<\delta$

But I did not get very far, is this the right definition to use? Could anyone point me in the right direction of how to complete this proof? Sorry if this isn't formatted well I'm new to LaTeX. Thanks

Answer 1

Since $\sin (x)$ is continuous all you have to do is check continuity at the points $x=\frac 1 n $ and $x=\frac 1 {n+1} $ For this use continuity of $\sin (x)$ and the fact that $\sin (n\pi)=\sin ((n+1)\pi)=0$.