Surely if you've reached contour integration, you've proved that a function of a complex variable being differentiable is equivalent to being analytic which is in turn equivalent to being infinitely differentiable (these statements are NOT equivalent in general for functions of a real variable). Furthermore, assuming D is simply connected, you know that an analytic function on D possesses an anti-derivative on D (Cauchy Integral Formula). Finally, the Cauchy-Riemann equations give NECESSARY conditions for which a function is analytic (i.e., if they fail, the function is NOT analytic; if they hold, it proves nothing). Under the additional assumption of continuity of first partial derivatives of some function, then the converse is true (i.e. they are sufficient).
I'm sure you can reformulate the above into a decisive proof. I suggest reviewing an getting completely straight the hypotheses and implications of the basic theorems of complex analysis right away (Cauchy-Riemann, Cauchy Integral, Cauchy-Goursat, Morera, analyticity, etc.), otherwise you're doomed to be constantly confused.
BTW, every integral (in the 1-dimensional case) is a path integral; when you integrate on the real line, you are still integrating on a path. The various forms of the fundamental theorem of calculus (Stoke's Theorem) show that under certain conditions, an integral over a "closed" manifold can be replaced by another integral on the boundary of said manifold. As it applies to your course, a path integral can be replaced by integrating the anti-derivative on the boundary of the path (i.e. the end-points) which then reduces to the difference of the anti-derivative (or "potential") function. If the contour is closed, you can again apply the anti-derivative on the "boundary" and obtain zero since you will be taking the difference of the anti-derivative at the same point (this is the essence of Morera's Theorem; the converse to Cauchy's Theorem). Green's Theorem also applies for closed contours (again, with certain assumptions on the domain, f, etc.).