Ultradifferentiable functions on smooth plane curves

The regularity, in the sense of ultradifferentiability, of real functions of two variables is determined in terms of the regularity of their restrictions to a given family of smooth plane curves. The special case of line segments reduces to the main result in Proc. Amer. Math. Soc. 127 (1999) 2099-2104]. As a consequence, the Bochnak-Siciak theorem on real analyticity is obtained. A formal analog of one of the results provides a generalization of the two-variable case of the Abhyankar and Moh J. Reine Angew. Math. 241 (1970) 27-33] theorem.