A foundational analysis of functional calculus

We introduce the local functional calculus, a unifying framework for local functional operators built on arbitrary continuous functions. This framework subsumes many known functional, and fractional calculus operators (conformable, non-conformable) and, more generally, any local derivative or integrator with a dependent scaling. As a special case, we develop the ocal Euler fractional calculus with the exponential kernel, which yields a particularly elegant calculus. For both the general functional and the Euler fractional cases we prove the fundamental properties: linearity, product rule, continuity, Rolle’s theorem, the mean value theorem, and the fundamental theorem of calculus. Crucially, we derive the correct chain rule, which rectifies an error present in earlier literature on generalised conformable derivatives. The local functional calculus thus provides a solid, self‑contained foundation for a wide class of local functional calculus operators, extending far beyond the traditional fractional setting.

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