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202212311431 Problem 1.4

Let g:\Omega\to\mathbb{R} be such that \displaystyle{\frac{\partial g}{\partial x}} and \displaystyle{\frac{\partial g}{\partial y}} exist at (x_0,y_0)\in\Omega, and suppose that one of these partials exists in a neighbourhood of (x_0,y_0) and is continuous at (x_0,y_0). Show that g is real-differentiable at (x_0,y_0).

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Roughwork.

Granted that

\begin{aligned} w=f(z)&=u(x,y)+iv(x,y) \\ \textrm{s.t. }A & =u_0+iv_0 \\ z_0 & =x_0+iy_0 \\ \end{aligned}

the concepts of existence of limit

\begin{aligned} &\qquad\enspace \lim_{z\to z_0}f(z) = A \\ & \Longleftrightarrow \begin{cases} \lim_{(x,y)\to (x_0,y_0)}u(x,y)=u_0 \\ \lim_{(x,y)\to (x_0,y_0)}v(x,y)=v_0 \\ \end{cases} \textrm{exist} \end{aligned}

and continuity at a point

\begin{aligned} &\qquad\enspace w=f(z)\textrm{ continuous at }z_0 \\ &\Longleftrightarrow \textrm{ both }u(x,y)\textrm{ and }v(x,y)\textrm{ continuous at }z_0 \\ \end{aligned}

are held by definition.

This problem is not to be attempted.