Holomorphy of an arbitrary approximately holomorphic mapping from a plane domain into the plane

A one-to-one mapping $f\colon D\to C$, with $D$ a domain in the complex plane $C$, is considered without the assumption that is continuous in $D$. It is proved that if the mapping $f\colon D\to C$ has a finite approximate derivative at each point of $D$, then it is holomorphic in $D$.