Baire irresolvable spaces and lifting for a layered ideal

We show the consistency (modulo reasonable large cardinals) of the existence of a topological space of power א₁ with no isolated points such that any real values function on it has a point of continuity. This is deduced from the following (by Kunen, Szymanski and Tall). We prove that if 2^λ = λ⁺, l is a λ-complete ideal on a regular λ which is layered, then the natural homomorphism from P(λ) to P(λ)/I (as Boolean algebras) can be lifted, i.e., there is a homomorphism h from P(λ) into itself with kernel I such that for every A ⊆ λ we have ≡ h(A) (mod l).