Let \begin{equation}h(x,y)= \begin{cases} 0, x \leq 0 \\ x(10-5y), x>0 \end{cases}\end{equation} be a piecewise smooth funcion, that is continuous everywhere, and smooth everywhere except at $(0,y)$.
My idea is to smooth out this "corner line" possibly by extracting a small domain around it, a stripe of width $2\delta$, for $\delta$ small but positive, and replace this with a function $g$ such that $f$ and $g$ can be glued together into a new function $H$ that is everywhere smooth.
It seems that all similar questions concern single variable functions, so this one seems trickier to me.
Thank you in advance!
Best Answer
I think the following works:
$$\begin{equation}h(x,y)= \begin{cases} 0,\ x \leq -\delta \\ \frac{(10-5y)(x+\delta)^2}{4\delta},\ |x|<\delta \\ x(10-5y),\ x\ge\delta \end{cases}\end{equation}$$
I simply treated $y$ as a constant, turning the function into a single-variable piecewise linear function. The two straight lines can be connected by a parabola, giving the result above.