390 Chapter^6in D. By improving on the last two results Menchoff was able to obtain
minimal conditions for analyticity. For this and other Menchoff theorems
we refer the reader to his monograph (83].
In the theorems mentioned above the condition that f be univalent
is essential. Trokhimchuk [121] has obtained several generalizations of
the theorems of Bohr and Menchoff to the cas~ of arbitrary continuous
mappings. His method is based on the consideration of the so-called mono-
geneity sets of continuous fnnctions together with the theory of interior
mappings and, in particular, on the following result: If f is continuous in
the domain D and analytic outside some perfect nowhere dense set PC D,
and if f is nonconstant on P, then there is a point zo E P such that f is
univalent in some neighborhood of zo.
Following N. N. Luzin, the monogeneity set of the continuous function
w = J(z) at a point z E D is defined to be the set
where Me is the set of values of the increment ratio ~w/ ~z for all ~z such
that 0 < l~zl :::; c:. It can be shown that at almost all points of D the
monogeneity set off is either a point, a circle, or the entire plane..
In a 1973 note, A. Pacquement (89] proposes to characterize the mono-
geneity in a class of not necessarily continuous functions. He defined f to be
generalized monogenic in a simply connected domain D if f is (1) general-
ized absolutely continuous, (2) approximately continuous, (3) of Baire class
1, and ( 4,) approximately differentiable almost everywhere in D. Then it
follows that the Cauchy-Riemann equations are satisfied almost everywhere
in D, the derivatives being in the approximate sense. For the appropriate
definitions of the concepts above, see S. Saks [107].
6.24 GENERALIZED ANALYTIC FUNCTIONS
Let f = u +iv E 'D(A), A open. If the components u and v are required
to satisfy a system of two partial differential equations of the first order
that are similar or that may reduce to the Cauchy-Riemann equations for
a particular choice of the functions or parameters involved, there results
a subclass of 'D(A) which in most cases has properties analogous to those
of the more restricted class of the analytic functions. This idea has been
explored by several authors, leading to a number of theories of generalized
analytic functions. Each theory has applications to certain types of partial
differential equations, or to systems of such equations, as well as to those