Inverse of continuous function on unbounded domain 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?
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.
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.