This is exercise (c) on page 6 of ``Elementary differential topology" by Munkres.
Find an open subset $U$ of $\mathbb R^2$ and a $C^1$ map $f : A \to \mathbb R$ ($A = \overline U$) such that $Df(x)$ ($x \in A$) depends on the extension of $f$ to a neighbourhood of $A$ in $\mathbb R^2$ which is chosen.
By definition $f$ is of class $C^1$ if it may be extended to a some neighbourhood of $A$ so that the extended function is of class $C^1$.