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

Each interval $(a,b)$ is open in the usual metric: if $x\in (a,b)$, the ball with radius $\min (x-a,b-x)/2$ is contained in it. Conversely, if $U$ is open and $x\in U$, the metric definition supplies $r>0$ with $(x-r,x+r)\subseteq U$. Thus every open set is a union of open intervals, which proves they form a basis.

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.