Let $D$ be a star-domain, and let $f$ be a continuous complex-valued function on $D$. Suppose that for every triangle $T$ contained in $D$ we have

$$\int_{\partial T} f(z) d z=0$$

Show that $f$ has an antiderivative on $D$.

If we assume instead that $D$ is a domain (not necessarily a star-domain), does this conclusion still hold? Briefly justify your answer.