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.

`eom.springer.de`

is broken, but the article can now be found at encyclopediaofmath.org/wiki/Approximate_derivative. $\endgroup$