On the algebra of functions $\mathcal {C}^k$-extendable for each $k$ finite

Abstract:

For each positive integer $l$ we construct a $\mathcal C^l$-function of one real variable, the graph $\Gamma$ of which has the following property: there exists a real function on $\Gamma$ which is $\mathcal C^k$-extendable to $\mathbb {R}^2$, for each $k$ finite, but it is not $\mathcal C^{\infty }$-extendable.