Sets with distinct sums of pairs, long arithmetic progressions, and continuous mappings

We show that if $\phi \text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ is a continuous mapping and the set of nonlinearity of $\phi$ has nonzero Lebesgue measure, then $\phi$ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.