On the topology of pointwise convergence on the ... - CARMA

Pro<strong>of</strong>: In order to obtain a contradiction let us supposethat f does not have a dense set <strong>of</strong> points <strong>of</strong> continuity withrespect to <strong>the</strong> norm <strong>topology</strong> on X. Since A is a Baire spacethis implies that for some ε > 0 <strong>the</strong> open set:O ε := ⋃ {U ⊆ A : U is open and ‖ · ‖-diam[f(U)] ≤ 2ε}is not dense in A. That is, <strong>the</strong>re exists a nonempty opensubset W <strong>of</strong> A such that W ∩ O ε = ∅. For each x ∈ A, letF x := {y ∈ A : ‖f(y) − f(x)‖ > ε}.Then x ∈ F x for each x ∈ W. Moreover, since A has countabletightness, for each x ∈ W, <strong>the</strong>re exists a countablesubset C x <strong>of</strong> F x such that x ∈ C x .