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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.

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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 $0$.

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