Take the tour ×  In real analysis one can define something known
as the approximative derivative of a function. See here eg
Roughly speaking one asks that the limit of the difference
quotient exists as long as h goes to zero while only taking
values in some subset that is sufficiently dense.  Does anyone
know if this concept has been studied for complex-valued valued
functions of a complex variable? The basic definition should go
through without problem so it should make sense to speak of an
approximately holomorphic function as one that has an
approximate complex derivative at every point of some open set.
It would be interesting how much of the classical complex
analysis one could generalize. Even knowing whether there exists
functions that are approximately holomorphic but not holomorphic
in the normal sense would be interesting.  share|improve this
question add comment  3 Answers  up vote 5 down vote accepted
Men'shov proved in 1936 that if $f\colon D\to\mathbb C$ is
continuous and approximately differen