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

We show that if $\varphi \colon \mathbb R\rightarrow\mathbb R$ is a continuous mapping and the set of nonlinearity of $\varphi$ has nonzero Lebesgue measure, then $\varphi$ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.