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?

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 $f(w)=1/(2\pi i)\int _ {\partial D} f(z) /(z-w) dz$.