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1. $f:\mathbb R^2\to \mathbb R$ is $C^\infty$.

2. $f:\mathbb R^2\to \mathbb R$ is $C^\infty$ along each $C^\infty$-curve $c:\mathbb R\to \mathbb R^2$; i.e., $f\circ c$ is $C^\infty$ for each such $c$.

Equivalence was proved only in 1979 by Jan Boman.