I was reviewing some old problems and I came across a problem which I found interesting. The problem although not very difficult was neat because it was something that was implicitly implied often but never proven, the question was

My solution was simple, I conjectured that the homeomorphism was linear and found that the function $\displaystyle f(x)=\frac{d-c}{b-a}x+\frac{bc-ad}{b-a}$ satisfies the correct conditions. The book has just answers and validated the above function, but I see a problem that I wish someone else could help me with. Don't we need connectivity for this homeomorphism to apply? Otherwise how can we assume that all intermediate values are taken?Prove that any open interval in $\displaystyle \mathbb{R}$ is homeomorphic to any other interval in $\displaystyle \mathbb{R}$

Thanks for any help

Alex