SCHWEITZER 2010 Problem 6

Is there a continuous function $f: \mathbb {R} ^ {2} \rightarrow \mathbb {R}$ for every $d \in \mathbb {R}$ we have $g_{m,d}(x) = f (x, m x + d)$ is strictly monotonic on $\mathbb {R}$ if $m \ge 0,$ and not monotone on any non-empty open interval if $m <0?$