We work in $\mathbb{R}^2$. Given a non-degenerate triangle $\triangle ABC$ and an interior point $P$, we specify the value of a function, value of its gradient at the $A,B,C$, and we specify the normal derivative at the midpoints of the vertices. I have shown that for every additional specification of the value and gradient of the function at $P$ there is a unique piecewise (three pieces being $\triangle ABP, \triangle ACP, \triangle BCP$) at most degree 3 on each piece function $Z$ which interpolates correctly.
Furthermore, I have shown that $Z$ is continuous.
The last bit I need to show is that there is a choice of what the value and gradient should be at $P$ which makes the corresponding $Z$ have a continuous gradient.
Any suggestions for me? In order to prove the first half I used the fact that $\dim \mathcal{P}_3[x,y] = \dim \mathbb{R}^{10}$ and used the fact that for linear maps on vector spaces of the same dimension we have injective iff surjective.