The set of differentiability of an extension from half-plane to the plane

Question

This question is related to: Differentiability: Partially Defined Functions

Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$.

Are there some that admit no extension differentiable in more than one boundary point? $$a\in\partial\mathbb{H}^2:\quad f_E:U_a\to\mathbb{R}$$

Answer 1

The function $f(x,y)=y\sqrt{x}$, originally defined in the right halfplane, can be extended as a function differentiable at $(0,0)$, but no extension is differentiable at other points of the $y$-axis.