1
$\begingroup$

Consider the following property: "Let $A$ be a closed bounded set of $\mathbb{R}$. Suppose that $f:A \rightarrow \mathbb{R}$ is a continuous injective function. Then $f^{-1}:f(A) \rightarrow A$ is also continuous."

I am looking for an example of a continuous injective function $f:A \rightarrow \mathbb{R}$ on a closed but unbounded set $A$ of $\mathbb{R}$ for which $f^{-1}$ is not continuous, in order to get an idea of how important the conditions for this property are. I can't think of a good example though. Any thoughts?

$\endgroup$
-1
$\begingroup$

I don't see anyway you will get such examples. Coz it's beyond the theorem. Note that if $A$ is compact and $f$ is continuous then $f(A)$ will be also compact. Moreover is it possible if $[a,b]$ is closed and not bounded? Whereas $a-\epsilon$ and $b+\epsilon$ are it's lower and upper bounds. Remember Every continuous function in $[a,b]$ is bounded.

$\endgroup$
  • $\begingroup$ Welcome to math.stackexchange! And thanks for your input! I would however like to point out that we use mathjax here. In essence is simply allows you to write in $\LaTeX$, using a dollar sign on either side of your code. $\endgroup$ – Mitchell Faas Jun 24 '17 at 13:30
  • $\begingroup$ This doesn't answer the question at all. $\endgroup$ – zhw. Jun 25 '17 at 6:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.