Using iterated uniform local completion, we introduce a notion of continuous (Formula presented.) functions on locally closed subsets of reduced complex spaces, generalising both holomorphic functions and (Formula presented.) functions on (Formula presented.) submanifolds. Under additional assumptions of set-theoretical weak pseudo-concavity, we prove optimal maximum modulus principles for these functions, extending classical results for holomorphic functions and ordinary (Formula presented.) functions. Restricting to real submanifolds (possibly with (Formula presented.) singularities) of complex manifolds, we generalise results on holomorphic extension to full neighbourhoods known before only for (Formula presented.) submanifolds. The article is concluded by a study of (Formula presented.) singularities and explicit constructions of submanifolds on which the extension results are valid.