Can we use the increment theorem for proving that functions with bounded partial derivatives is continuous ?
(There is one question similar to this asked in 2013.But I want to know what is wrong with this proof)
$\delta z= f_{x}\delta x+f_{y}\delta y+\epsilon_{1}\delta x+\epsilon_{2}\delta y$ where $\epsilon_{1} $ $\epsilon_{2}$ tends to $0 $ as $\delta x,\delta y$ tends to $0$ But $\delta z =f(x+\delta x,y+\delta y)-f(x,y)$ Applying boundedness of partial derivatives $f_{x} $and $ f_{y}$ we get as $limit f(x+\delta x,y+\delta y)=f(x,y) $ as $\delta x,\delta y$ tends to $0$.
You do not know $f$ is differentiable just because it has partial derivatives. Of course, differentiability always implies continuity.
EDIT: To clarify, your stipulation that $\epsilon_i\to 0$ requires that $f$ be differentiable. We don't have that.