1

Let $f:A \rightarrow B$ be a bijection, where $A$ and $B$ are subsets of $\mathbb{R}$. Prove that if $f$ is increasing on $A$, then $f^{-1}$ is increasing on $B$.

I have an idea of the picture of how this is true, but I don't know how to prove this with words. Any help is greatly appreciated.

hawk2015
  • 117
  • what's your idea. – uniquesolution Sep 23 '15 at 03:25
  • "Any help is greatly appreciated" yeah sure. It seems that you don't even look at the answers you receive (if you think they were helpful you must select the best answer or at least say thanks or something). – CIJ Oct 06 '15 at 06:47

1 Answers1

1

Hint: Suppose $x<y$ in $B$. First use the fact that $f$ is surjective, and obtain a fact about the preimages of $x$ and $y$ in $A$.

Another way (assuming differentiable): $f(f^{-1}(x))=x$. Take the derivative with respect to $x$ of both sides, and use the fact that $f'(*)>0$

David P
  • 12,320