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. 
EDIT: It was 1967, sorry for being careless.

EDIT: Using "general abstract nonsense" and functional analysis, one push this result from $\mathbb R^2$ to Frechet spaces. Beyond Frechet spaces, the notions start to divergence. 
Analysis based on (2) is called convenient analysis, since it leads to a diffeomorphism
$$C^\infty(U,C^\infty(V,W)) \cong C^\infty(U\times V, W)$$
and a monoidally closed category.

See:

A.Frölicher and A.Kriegl:
Linear spaces and differentiation theory.
John Wiley & Sons Ltd., Chichester, 1988.

Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997.
[(pdf)][1]
 


  [1]: http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf