# Proof of Discontinuity: $f:\mathbb{R}\to \mathbb{R}, x\mapsto f(x)=\sin(1/x)$

Prove the Discontinuity of $$f:\mathbb{R}\to \mathbb{R}: x\mapsto f(x)=\begin{cases} \sin\left( \frac{1}{x}\right) \quad \quad x\neq 0\\0 \quad \quad \quad \quad x=0\end{cases}$$.

My Proof:

Since $$\forall x\neq 0$$ $$f$$ is a composition of continunous functions, it is continuous. It is not continuous in $$x_0=0$$ since for $$(x_n)_{n\in \mathbb{N}}$$ with $$x_n:=\frac{1}{n}\to 0$$ since:$$\lim\limits_{n\to\infty}f(x_n)=\lim\limits_{n\to\infty}\sin(n)$$ This limit does not exist.

While it is true that $$\lim_n \sin \, n$$ does not exist it is not very easy to prove. Instead of this you can take $$x_n=\frac 2 {(2n-1)\pi}$$ to see easily that $$f$$ is not continuous at $$0$$.