Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D\_1$ and $D\_2$.


Let *f* : D→ℂ be a continuous function that is holomorphic on the interiors of $D\_1$ and $D\_2$.  
 Is *f* then necessarily holomorphic?




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PS: If he path C is sufficiently smooth (so that ∫C *f*(*z*) d*z* makes sense), then f is necessarily holomorphic, as it is then given by Cauchy's formula $f(w)=1/(2\pi i)\int \_ {\partial D} f(z) /(z-w) dz$.



Continuous + holomorphic on a dense open => holomorphic?
[Denjoy](http://www.numdam.org/item?id=BSMF_1932__60__27_0) makes a detailed study of this question, and in particular constructs counterexamples where the curve C is the graph of a continuous function. Apparently, the construction works for curves which are 'very' non rectifiable, i.e., the local variation is infinite of a suitably high order at each point.


