Let $a$ and $b$ be real numbers such that $a<b$ and $(\mathbb R, \tau)$ a topological space generated by the Euclidean metric. I need to show that the open interval $(a,b)$ with the subspace topology is homeomorphic to $(\mathbb R, \tau)$.
Homeomorphic means that there is a function Continuous, one-to-one, in surjection, and having a continuous inverse between the two spaces.
I found the function $$ f(x)=tan(\pi/(b-a)x-pi(a+b)/(2(b-a))$$ that is $f:(a,b)->\mathbb R$
How do I show that this function is continuous?
I think that since its derivative exists and is continuous then my $f(X)$ is continuous. I think it is it discontinuous only at $a$ and $b$?