Perhaps the Intermediate Value Theorem

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 one may derive a contradiction from the assumption that I0 coincides with F . We show that Brouwer’s Continuity Principle also enables us to define uncountably many subsets G of F with the property I0 ⊆ G ⊂ (I0)¬¬.