Zulfiability: Convex-Like Functions under Zulfia Decomposition Mapping

Convexity and invexity are classical convex-like structures based on directionalbehavior. Convexity uses the canonical direction x − u, while invexity replaces itwith a generalized mapping η(x, u).In this paper, we introduce Zulfia mappingZ(x, u) = η(x, u) + ∆(x, u),which decomposes generalized direction into directional and deviation components.We establish an Unavoidable Decomposition Theorem showing that any general-ized direction admits a direction–deviation representation. This provides a compactreference framework for Zulfia convexity and Zulfia invexity, with classical struc-tures recovered as limiting cases. We call these functions as Zulfiable functions orZulfiability.