On the strength of sufficient condition for complex differentiability

Question

Let $f:A \to \Bbb C$, $A \subseteq \Bbb C$, be a function with the form $f=u+iv$, The very familiar sufficient condition for differentiability is the continuity of partial derivatives of $u$ and $v$ in addition with Cauchy-Riemann equations.

But,

i. Considering $f$ as a function from a subset $A$ of $\mathbb R^2$ to $\Bbb R^2$, we need only the continuity of one of the partial derivatives of $u$ and $v$ for the differentiability (Apostol, Mathematical Analysis, Theorem 12.11).

ii. Also, $f$ is complex differentiable if and only if $f$ is real differentiable in addition to Cauchy-Riemann equations.

From the above premises (i and ii), I can conclude that the continuity of one of the partial derivatives of $u$ and $v$ besides Cauchy-Riemann equations is sufficient for the complex differentiability of $f$.

I feel something is missing since the conventional sufficient condition is slightly different.

Any analytical error committed in the deduction?

Answer 1

You are right, the "traditional" theorem is

• If $f$ is continuously partially differentiable at $z_0$ and satisfies the Cauchy-Riemann equations at $z_0$, then $f$ is complex differentiable at $z_0$.

This is a consequence of the following two ingredients:

1. $f$ is complex differentiable at $z_0$ if and only if it is real differentiable at $z_0$ and satisfies the Cauchy-Riemann equations at $z_0$.

2. If $f$ is continuously partially differentiable at $z_0$, then it is real differentiable at $z_0$.

As you noticed, the assumption of 2. can be weakened:

3. If $f$ is partially differentiable at $z_0$ and one of the partial derivatives $\partial f/\partial x, \partial f/\partial y$ exists in a neigborhood of $z_0$ and is continuous at $z_0$, then $f$ is real differentiable at $z_0$.

This allows to formulate a variant of the above theorem with a slightly weaker assumption (as you have done in your question).

However, the weaker assumption in 3. is not really relevant. In practice we are interested in holomorphic functions $f : U \to \mathbb C$ on a region $U \subset \mathbb C$, i.e. functions which are complex differentiable at all points of $U$. Even without using 3. we get the following theorem:

• Let $f = u + iv$ be partially differentiable in all points of $U$ (that is, $u_x = \partial u/\partial x, u_y = \partial u/\partial y, v_x = \partial v/\partial x, v_y = \partial v/\partial y$ exist in all points of $U$) such that the Cauchy-Riemann equations are satisfied at all points of $U$. Assume that in each point $z_0 \in U$ one of the following conditions is satisfied:
  (a) $u_x, v_x$ are continuous at $z_0$.
  (b) $u_y, v_y$ are continuous at $z_0$.
  (c) $u_x, u_y$ are continuous at $z_0$.
  (d) $v_x, v_y$ are continuous at $z_0$.
  Then $f$ is holomorphic.

In fact, under the above assumptions the Cauchy-Riemann equations imply that both $u,v$ have continuous partial derivatives at each point of $U$ - and this is the traditional assumption.