Exact real number computations relative to hereditarily total functionals

We show that the continuous existential quantifier ∃ω is not definable in Escardó’s Real-PCF from all functionals equivalent to a given total one in a uniform way. We further prove that relative to any total functional of type (I → I) → I which gives the maximum-value for any total input, we may, given a computable, total functional Φ of type (R → R) → R find a Real-PCF definable total Ψ equivalent to Φ.