I have been taking a course on topology this semester and I need help in this particular exercise.
Prove that the function $f: (0,1)\to \mathbb{R}$ defined by $f(x)= \sin(1/x)$ cannot be extended to a continuous function with domain $[0,1]$.
We have studied 1 point compactifications but here 2 points are being added. I am unable to think of which result to use and would very much appreciate hints.
I am doing masters level course but first course.
$1$ is irrelevant, there also is no extension to $[0,1)$, e.g.
Consider the sequences $a_n = \frac{1}{\pi n}$ and $b_n = \frac{1}{2\pi n + \frac{\pi}{2}}$ in $(0,1)$.