In Domain Realizability, not all Functionals on C[-1, 1] are Continuous

In this note we exhibit a continuity principle for real-valued functions on C[−1, 1] that is not validated by realizability over domains although it is validated by Kleene’s functional realizability corresponding to Weihrauch’s theory of type 2 effectivity. Mathematics Subject Classification: 03F50, 03F60, 18C50, 06B35.