A piecewise function $F(X_1,X_2)$ is defined on domain $D=\{(X_1,X_2)|X_2\ge X_1\ge 0\}$.
If $X_2> X_1+K_1$
$$F(X_1,X_2)=\begin{cases} \varphi(\hat Y)+h_2(X_2-\hat Y), \hat Y-K_1\le X_1\le \hat Y\\ \varphi(X_1+K_1)+h_2(X_2-X_1-K_1), X_1\le \hat Y-K_1\\ \varphi(X_1)+h_2(X_2-X_1), X_1\ge \hat Y\\ \end{cases}$$
If $X_2\le X_1+K_1$
$$F(X_1,X_2)=\begin{cases} \varphi(\hat Y)+h_2(X_2-\hat Y), X_1\le \hat Y\le X_2\\ \varphi(X_2), X_2\le \hat Y\\ \varphi(X_1)+h_2(X_2-X_1), X_1\ge \hat Y\\ \end{cases}$$ where $K_1$ is a constant, $\hat Y$ is the unconstrained minimizer of $\varphi(Y)+h_2(X_2-Y)$, $\varphi(Y)$ is a convex function.
One can see that $F$ is continuous on $D$, but its 1st order, 2nd order partial derivatives are not. However, $F''_{ij}\ge0$, so, is $F$ convex? How about general continuous functions $F$ with discontinuous 1st order, 2nd order partial derivatives and $F''_{ij}\ge0$?
