Solution: We will give a homeomorphism between these two intervals as follows. Map $a$ to $c$ and $b$ to $d$. The map will be the line joining $(a,c)$ to $(b,d)$. The equation of the line will be \[ y(x) = \left( \frac{c-d}{a-b} \right) (x-a) + c. \] So, we define the map \[ \varphi : (a,b) \to (c,d), \quad \varphi (x) = \left( \frac{c-d}{a-b} \right) (x-a) + c. \] Note that, $\varphi $ maps into $(c,d)$. It is continuous as it is a linear polynomial. Also, it is clear that it is a bijection between $(a,b)$ into $(c,d)$.