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We show that if $\varphi \colon \mathbb R\rightarrow\mathbb R$ is acontinuous mapping and the set of nonlinearity of $\varphi$ has nonzeroLebesgue measure, then $\varphi$ maps bijectively a certain set that containsarbitrarily long arithmetic progressions onto a certain set with distinct sumsof pairs.

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