# Show that $\cos(1 / x)$ cannot be continuously extended to $0$

Can anyone help me to solve this problem?

Q: Show that $\cos(\frac{1}{x})$ cannot be continuously extended to $0$?

Notice: use this Theorem

Let $f:D\rightarrow \mathbb{R}$ and let $c\in D$. Then $f$ is continous at $c$ if and only if, whenever $X_n$ is a sequence in $D$ that converges to $c$, then $f(X_n)$ converges to $f(c)$.

I wish someone can solve this problem

Thanks

• What have you tried so far? Can you find a sequence of values $t_n$ with $t_n\to\infty$ and $\cos(t_n)$ having no limit? Can you see how you might use that sequence to solve your problem? – Steven Stadnicki Apr 8 '13 at 6:55
Consider the sequence $X_n = \frac{1}{\pi n}$.
• Then supposing $f$ could be continuously extended, and taking $c=0$ in the theorem, what can be deduced about the value $f(0)$. – Jeppe Stig Nielsen Apr 8 '13 at 7:03
Consider the sequences $$x_n = \dfrac1{2n \pi} \,\,\,\, \text{ and } x_n = \dfrac1{2 n \pi + \dfrac{\pi}2}$$ Both tend to zero. What happens to $\cos(1/x)$ along these sequences? (Recall that if a limit exists it has to be unique.)