Theorem
Consider the continuous function $f: (0,1]\mapsto\mathbb{R}$ defined by $f(x)=\sin(\frac{1}{x}).$ I have to answer the following question :
show that it is impossible to extend this
function to continuous function from
$[0,1]$ to $\mathbb R$
It is not clear to me what is the question asked me to show:
Is it asking me to show that $\sin(\frac{1}{x})$ is discontinuous ?
If not what is it asking me to do ,exactly?
This question is in the Compactness chapter.Either way best way to prove it
is to use proof by contradiction
PS:I thought of using Intermediate Value Theorem
Best Answer
The thing the question wants you to say is that it is not possible to define $f(0)$ which makes the function continuous on $[0,1]$.
To see that you define $f(0)=r$ say . for arbitrary $r\in\mathbb{R}$
You will get that for $\epsilon=\frac{1}{2}$. You have $\exists \,x$ such that
$|\sin(\frac{1}{x})-r|\geq \frac{1}{2}$ for all $\delta>0$ such that $0<|x|<\delta$.
So you see that no-matter how you define $f(0)$, you will end up with a discontinuity at $x=0$.