This paper consists of three parts: Starting with the field $mathbb{C}$ of complex numbers,the Banach space $mathcal{C}(V)$ of continuous, $mathbb{C}$-valued functions on a simply connected compact region $Vsubsetmathbb{C}$ is shown to decompose the topological direct sum of two complementary subspaces: a subspace of integrable and therefore analytic functions,and a subspace of un-integrable and anti-analytic functions. Introducing orientation as a central notion, orientation inversion (parity) turns out to be the complex conjugation, which maps integrable (w.r.t. positive orientation) to un-integrable functions, which are integrable w.r.t. negative orientation, and vice versa. Orientation allows the extension of complex analyticity to $mathbb{R}^2$, which ends part 1. Part 2 is devoted to the extension of analyticity to multi-dimensions.These results are then applied in part three to continuous mechanical dynamical systems, where it is shown that Hamilton-Jacoby theory yields the unrestricted integrability of any continuous, mechanical dynamical system of either parity and represents their solutions as geodesics of (integrated) action functions of positive/negative parity, i.e.: as fermionic and bosonic solutions.