I would like to prove for a function, $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$, at two different points, $(x_1, y_1)$ and $(x_2, y_2)$, the absolute difference between values of the function is bounded,
$$ |f(x_1,y_1) - f(x_2,y_2)| \leq |\partial_x f(x_c,y_c)|\ |x_1 - x_2| + |\partial_y f(x_d,y_d)|\ |y_1 - y_2| $$
where $\min(x_1,x_2) \leq x_c, x_d \leq \max(x_1, x_2)$ and $\min(y_1, y_2) \leq y_c,y_d \leq \max(y_1,y_2)$. It seems obvious to me, but I don't know how to prove it formally. If anyone could show the formal proof, that would be great. Thank you.
Hint: put $$g(t)=f(tx_1+(1-t)x_2,ty_1+(1-t)y_2)$$ You have $$g(1)-g(0)=f(x_1,y_1)-f(x_2,y_2)=g^{\prime}(u)$$ for an $u\in ]0,1[$.