  We introduce a notion of locally approximable continuous CR functions on
locally closed subsets of reduced complex spaces, generalizing both holomorphic
functions and CR functions on CR submanifolds. Under additional assumptions of
set-theoretical weak pseudoconcavity we prove optimal maximum modulus
principles for these functions. Restricting to real submanifolds (possibly with
CR singularities) of complexmanifolds, we generalize results on holomorphic
extension known for CR submanifolds.
