Modified IVT theorem in $\mathbb{R}^2$

Question

Poincaré–Miranda theorem in $\mathbb{R}^2$: $f,g:[-1,1]\times[-1,1]\to \mathbb{R}$ continuous if $f(-1,y)<0,f(1,y)>0, g(x,-1)<0$ and $g(x,1)>0$ then there exists $(x_0,y_0)$ such that $f(x_0,y_0)=g(x_0,y_0)=0$ my question:1) i am trying to modify $f(-1,y)=0, g(x,-1)=0$ instead $f(-1,y)<0, g(x,-1)<0$ and added condidion $\nabla f(-1,-1)>0, \nabla g(-1,-1)>0$ for example, can i obtained the same result? (a priori yes) 2) my second question is, if we change the domain to $\{[0,1]\times[0,1]\; \text{such that}\; (x,y)\leq\gamma\}$, $\gamma$ is the decreasing path for $(0,1)$ to $(1,0)$. the hypothesis becomes $f,g<0$ on $\gamma$ and $f(x,0)=0$ and $g(x,0)=0$+ $\nabla f(0,0),\nabla g(0,0)>0$. Thanks anyone for proving that.