On the holomorphy of function which has the derivatives with respect to some sets (in Russian)

Let $f\colon D\to \Bbb C$, where $D$ is a domain in the complex plane $\Bbb C$, be a one-to-one mapping which is not supposed to be continuous. Suppose that for each $z\in D$, $f(z)$ satisfies two requirements:
1) the finite derivative $f'$ exists with respect to some measurable set $E(z)$ such that $\lim\limits_{r\to 1} \frac{\text{m}({E}(z)\cap C(z,r))}{\pi r^2}>0$, where $C(z,r)=\{z':|z'-z| < r\}$;
2) $\varlimsup\limits_{z'\to z, z'\in E(z)}\bigl|\frac{f(z')-f(z)}{z'-z} \bigr|<\infty$ for some measurable set $E(z)$ such that $\lim\limits_{r\to 0} \frac{m(E(z)\cap C(z,r))}{\pi r^2}=b>\frac12$. Then it is proved that function $f$ is holomorphic.