a) See the explanation below. b) See the explanation below.
Work Step by Step
a) A differentialable function $z=f(x,y)$ is defined as the linear approximation and continuous on $(a,b)$ when it is near to $(x,y)$. This can be described as: $\triangle z=f_x(a,b)\triangle x+f_x(a,b) \triangle y+\epsilon_1 \delta x+\epsilon_2 \triangle y$ Here $\epsilon_1,\epsilon_2$ approaches to $0$ when $(\triangle x,\triangle y) $ approaches to $0$. b) This means that the first partial derivatives of the function $f(x,y)$ , that is, $f_z, f_y$ should be continuous and exist near the point $(a,b)$