Let $B^+$ be the open half ball $\{(x,y)\in \mathbb{R}^2 |x^2+y^2<1,y>0\}$. Assume that a function $f\in C^2(B^+)$ is $f=0$ on the $x$-axis such that $f,\partial_{x} f\partial_{y} f,\partial_{x^2} f, \partial_{xy} f,\partial_{x^2} f$ are uniformly continuous (thus they all continuously extends to the boundary).
Set $g(x)\in C(B)$ as $f(x,y)$ if $y \ge0$ and $-f(x,-y)$ is $y<0$ for $(x,y) \in B=\{(x,y)\in \mathbb{R}^2 |x^2+y^2<1\}$. How can we prove that $f\in C^2(B)$?