Suppose $f(x,\,y)\in C^1(\mathbb R^2)$. If $\displaystyle\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$ and $f(x,\,0)>0$ for all $x\in\mathbb R$. Prove that $$ f(x,\,y)>0\quad\text{ for all $(x,\,y)\in\mathbb R^2$}. $$
I find that for all $a,\,b,\,c,\,d\in\mathbb R$, there exists $(\xi,\,\eta)\in\mathbb R^2$ such that $$ f(a,\,b)-f(c,\,d)=(a-c)\frac{\partial f}{\partial x}(\xi,\,\eta)+(b-d)\frac{\partial f}{\partial y}(\xi,\,\eta). $$
Suppose $f(x_0,\,y_0)$ for some $(x_0,\,y_0)$, then what will happen? How the condition $\displaystyle\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$ works to ensure the conculsion?
