New differentiability criteria for complex-valued functions (Q1905095)

The author deals with the generalization of the well-known theorem which implies that a function of two variables \(f(x, y)\) is differentiable at a point \((x, y)\) if both partials exist and are continuous at that point. For example, it can be shown, that for a Lipschitzian function (real or complex) the existence of the limits of the partials at some point through points of differentiability of \(f\) also imply differentiability of \(f\) at that point. Here no assumption is made on the existence of the partials. For arbitrary functions this assertion is not true. A counter example can serve a singular function \(\phi(x)\), \(x\in [a, b]\) and a point \(x_0\) where \(\phi'(x_0)= + \infty\) and defining \(f(z)= f(x+ iy)= \phi(x)\) at \(z= x_0+ iy\). Nevertheless, the author does generalize the above theorem for arbitrary complex functions \(f(z)\) replacing the partials by \(f_z\) and \(f_{\overline z}\), and replacing the limits by asymptotic limits. Let \(C_z\) denote the complex \(z\)-plane. A function is said to be monogenic (not necessarily in a domain) if it is complex differentiable. The main theorem of the paper is Theorem 1. Let \(f(z)\in \text{Lip}(D)\), \(D\subset C_z\), and let \(f_{\overline z}\to 0\) asymptotically as \(z\to z_0\). Then \(f(z)\) is monogenic at the point \(z_0\). As a consequence of Theorem 1, the author proves two additional theorems. Theorem 2. Let \(f(z)\in \text{Lip}(D)\), \(D\subset C_z\), and suppose that at a certain point \(z_0\in D\) there exists the asymptotic limit to \(f_z\) or \(f_{\overline z}\). Then \(f(z)\) is differentiable at \(z_0\). Furthermore, in the representation \(df= f_z dz+ f_{\overline z} d\overline z\) the corresponding coefficient equals to the above asymptotic limit. Theorem 3. Let \(f(z)\in \text{Lip}(D)\), \(D\subset C_z\). Assume that the monogenicity set \({\mathcal M}(f)\) is nowhere dense in the set \(E\subset D\) (not of the first category). Then the function \(f(z)\) is differentiable on the set \(E\), except possibly for a set of the first category, and the set \({\mathcal M}(f)\) consists only of circles or points. In both the Russian article and the English translation the expression \(f_{\overline z}\) is misprinted as \(f_z\) in the statement of the main theorem.