If $f:R\to \{0,1\}$ is a continuous surjection map then $f^{-1}\left(0\right)\cap f^{-1}\left(1\right)$ is:

since $f:R\to \{0,1\}$
domain = R and Range = {0, 1} 
Since each element of domains set have unique image either 0 or 1, 
Therefore pre images of 0 and 1are different elements
$\therefore f^{-1}\left(0\right)\cap f^{-1}\left(1\right)=\varphi$