Condition for closed contour integral to equal 0

Question

When you get a simple closed contour integral of the form:

$$\oint_C f(z)dz$$

can the integral equal $0$ even if it's not holomorphic at an interior point?

Answer 1

Yes, consider $$\int_{C(0,1)} \cos\left(\frac{1}{z}\right)dz.$$ The function is not holomorphic at $0$ but the residue is equal to $0$, hence the integral too.

Answer 2

Yes, a simple example would be $$ \int_{|z|=1} \frac{1}{z^2}\,dz. $$