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.