Discussion
Let f be continuous in an open set D, then \(\int_{C} f(z)\, dz = 0\) for each piecewise differentiable closed curve C in D if and only if :
Answer: D
The keyed condition is f is an analytic function on D.
The 'only if' direction is Morera's theorem: if f is continuous on an open set D and its integral over every closed curve in D is 0, then f has a primitive F locally (integrate along paths from a fixed point), F is analytic, and so its derivative f is analytic too.
The 'if' direction is Cauchy's theorem. Strictly it needs D to be simply connected: on the punctured plane, \(1/z\) is analytic but its integral around the unit circle is \(2\pi i\). The question follows the textbook pairing of Morera and Cauchy, under which analyticity is the characterising property.
The other conditions are too narrow or irrelevant. The zero function and polynomials do have zero integrals, but so does every analytic function such as \(e^{z}\), so neither is necessary. Periodicity is neither necessary nor sufficient: \(z^{2}\) is not periodic yet integrates to zero, while \(\sin(\text{Re}\, z)\) is periodic and continuous but its integral around the unit circle is \(i\int_{0}^{2\pi} \sin(\cos t)\cos t\, dt \ne 0\).
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