A little doubt related to Continuity

Question

I was reading a solution and there they have used a thing related to continuity which I don't understand and I quote here - this shows that either f(x)=1 or f(x)=-1 for each x..but continuity of f shows that f(x) is either identically equal to 1 or -1 ( here f is continuous)

WHY

I think this is enough details to clarify my doubts ..but if someone need more details then pls ask ..

Kindly pls someone help me... Thankyou very much

Answer 1

If $f$ is continuous and its domain $D$ is an interval, then, if there is a $x\in D$ such that $f(x)=1$ and if there is a $y\in D$ such that $f(y)=-1$ them by the intermediate value theorem, for each $c\in[-1,1]$, there is some $z\in D$ such that $f(z)=c$. So, since $f$ only take the values $\pm1$, $f$ must always take the value $1$ or only take the value $-1$.

Answer 2

I think we can prove this without directly using IVT by bisecting intervals.

Suppose $f$ is not constant, so there are points $a,b$ where $f(a) = -1$ and $f(b)=1$. Let $I_0=[a,b]$ and let $c$ be the mid-point of $I_0$. If $f(c) = 1$ then let $I_1=[a,c]$, otherwise let $I_1=[c,b]$. So we now have an interval $I_1$ that is half the size of $I_0$ and contains at least one point where $f(x)=-1$ and at least one point where $f(x)=1$. By defining $I_2, I_3, \dots$ in the same way we have a series of intervals where $I_n \subset I_{n-1}$, $|I_n|= \frac 1 2 |I_{n-1}|$ and each $I_n$ contains at least one point where $f(x)=-1$ and at least one point where $f(x)=1$.

By compactness of the real line there is a limit point $l$ that is common to all of the $I_n$. But given any $\delta > 0$ we can find an $I_n$ such that $|I_n| < \delta$. Since $I_n$ contains at least one point where $f(x)=-1$ and at least one point where $f(x)=1$, $\displaystyle \lim_{x \rightarrow l} f(x)$ does not exist, so $f$ is not continuous at $l$.

Therefore if $f$ is continuous everywhere then it must be constant.

My analysis lecturer called this method "lion hunting". If you know the lion is in the jungle but you don't know where then you keep bisecting the jungle and confining the lion to a smaller and smaller space.