Inverse function and continuity

[phymatter]

if a continious function is monotoniously increasing in an interval , is it necessary that its inverse will also increase monotoniously in that interval?

 

[dimitri151]

Wow this is a cool question. I think so.
You have: a<b then f(a)<=f(b).
You also have:f^-1(f(x))=x
You're wondering: if A<B is f^-1(A)<=f^-1(B)

I think it might actually be strictly monotone increasing for the inverse. I'll have to think about it some more tomorrow.

 

[HallsofIvy]

You don't even need "continuous". Suppose f is monotonically increasing on [a, b] but that $f^{-1}(x)$ is not. Then there exist u, v, in [f(a), f(b)] such that u> v but $f^{-1}(u)< f^{-1}(v)$. Let $p= f^{-1}(u)$ and $q= f^{-1}(v)$. Then we have $p< q$ but $f(p)= u> v= f(q)$ contradicting the fact that f is increasing.