Past exam of the options I course of the University of Oxford / 2008 / E / E1 / c / ii / Solution

If $f$ is continuous, the sets in question are preimages of the open rays $(-\infty ,\lambda )$ and $(\lambda ,\infty )$, so they are open. Conversely, suppose all those sets are open. For every $a<b$,is open as the intersection of two open sets. Part (i) and the basis criterion of part (b) then prove $f$ is continuous.