Definitions and notations. \(D\) is a domain in the complex plane, \(\sigma= p-iq\in C(D)\), \(p(z)> 0\), \(\forall z\in D\), a function \(f=u+iv\) of class \(C^ 1(D)\) is called \(\sigma\)-analytic in \(D\) if it satisfies one of the three equivalent conditions: (1) \({{\partial f} \over {\partial \overline{z}}}+ \lambda(z) {{\partial \overline{f}} \over {\partial \overline{z}}} =0\), \(\lambda(z):= {{\sigma- 1} \over{\sigma+1}}\); (2) \({{\partial u} \over {\partial \overline{z}}}+i {{\partial v} \over {\partial \overline{z}}} =0\); (3) \(p {{\partial u} \over {\partial x}} +q {{\partial u}\over {\partial y}}- {{\partial v} \over {\partial x}} =0\), \(-q {{\partial u} \over {\partial x}} +p {{\partial u} \over {\partial x}} =0\). For \(\sigma\in C^ 1\), \(f\in C^ 0\), denote \[ t_ \sigma(z)= \exp \biggl[ \int {1\over {2p(z)}} {{d\sigma} \over {d\overline {z}}} (z) d\overline {z} \biggr], \] \(\Delta_ \sigma f(a,dz)= \sigma(a) [u(a+ \Delta z)- u(a)]+ i[v(a+ \Delta z)- v(a)]\), for \(a,a+\Delta z \in D\). A complex number \(\lambda\in \overline{C}\) is called a \(\sigma\)-derivative number of \(f\) at \(a\in D\) if there exists a sequence \(\{ z_ n\}_{n=1}^ \infty \in D\) tending to \(a\) such that \[ \lim_{n\to\infty} t(a) {{\Delta \sigma f(a, \Delta z_ n)} \over {\Delta z_ n}} =\lambda, \qquad \Delta z_ n:= z_ n -a. \tag{1} \] If the above limit exists it is called the \(\sigma\)-derivative of \(f\) at \(a\) and is denoted by \(\partial_ \sigma f(a)\). The function \(f\) is then called \(\sigma\)-monogenic at the point. The set of all \(\sigma\)- derivative numbers of \(f\) at \(a\) is called a set of \(\sigma\)-monogeneity of \(f\) at \(a\) and is denoted by \(M^ \sigma_ a(f)\). The limit in (1) exists independently of \(t(a)\) and this factor is introduced to make some properties of the \(\sigma\)-derivative close to the ordinary complex derivative in which case \(\sigma(a)= t(a)=1\). \(M^ \sigma_ a (f,\varepsilon)\) denotes the set of all values of the ratio \(\{t(a) {{\Delta \sigma f(a, \Delta z)} \over {\Delta x}}\}\), for all \(\Delta z\) such that \(a+ \Delta z\in D\), \(00} \overline {M}^ \sigma_ a (f,\varepsilon)\). (b) If \(f\) is real differentiable at \(a\in D\) then \(M^ \sigma_ a(f)\) is the circle \(C(f^ \sigma_ z (a), | f^ \sigma_{\overline{z}} (a)|)\). If \(f^ \sigma_{\overline{z}}(a) =0\) the set \(M^ \sigma_ a(f)\) reduces to a point and \(f\) is \(\sigma\)-monogenic at that point. (c) The main result of the paper makes use of the function \(f_ 0\) which is obtained as the solution of the system (1) out of the three equivalent systems in the introduction. The function \(\lambda\) is defined outside \(D\) as zero and the unique solution \(f_ 0\) is holomorphic outside \(\overline {D}\), \(f_ 0 (z)\sim z\), \(z\to\infty\). Theorem. With the previous notations let \(M\) be the set of all points \(a\in D\) at which the set \(M^ \sigma_ a(f)\) does not coincide with the extended complex plane. Then (ca) for any \(n,m= 1,2,\dots\) there exists compacta \(M_{nm}\) and complex numbers \(b_{nm}\) such that \(M_ n= \bigcup^ \infty_{m=1} M_{nm}\), and for any points \(a\in M_{nm}\) and \(z\in D\), \(| a- z|< {1\over m}\), the inequality \[ | f(a)- f(z)+ b_{nm} (f_ 0(a)- f_ 0(z))| \geq \delta_{nm} | z-a|; \] holds; (cb) the function \(f_{nm}(z)=f(z)+ b_{nm} f_ 0(z)\) is real differentiable almost at every point of the set \(M_{nm}\); (cc) the function \(f\) is real differentiable almost at every point of the set \(M\), and the set of \(\sigma\)-monogeneity \(M^ \sigma_ a (f)\) is either a point or a circle.