In the context of intuitionistic real analysis, we introduce the set F consisting of all continuous functions φ from [0, 1] to R such that φ(0) = 0 and φ(1) = 1. We let I0 be the set of all φ in F for which we may find x in [0, 1] such that φ(x) = 12 . It is well-known that there are functions in F that we can not prove to belong to I0, and that, with the help of Brouwer’s Continuity Principle ...