If $D$ is a convex domain, $\gamma$ is a rectifiable closed path and $f(z): D \to \mathbb{C}$ is continuous, then
$$\int_\gamma f(z) \mathrm{d}z = 0?$$
I don't think this is correct, since Cauchy's integral theorem needs $f$ to be holomorphic. But all the functions I tried so far that are continuous but not holomorphic ($\mathrm{Re}(z),|z|,\bar{z}$) seem to have this property. Can you give me an example of a continuous function that doesn't have this property?