TITLE: When does a function take an open interval into an open interval?
QUESTION [5 upvotes]: Suppose we consider a function $f(x)$ defined from an open interval say $(a,b)$ to some set $T$. When would the set $T$ be an open interval?

REPLY [2 votes]: The set $T$ would be an open interval if the function is also injective, that is, supposing that T is a subset of R and $f$ is continuous.