 Solution

If $f$ is continuous, the two sets are inverse images of the open rays $(-\infty ,\lambda )$ and $(\lambda ,\infty )$, so they are open. Conversely, suppose both families of inverse images are open. For any interval $(a,b)$,which is open. By part (c)(i) and the basis of a topology criterion from part (b), all inverse images of open sets are open. Hence $f$ is continuous.