How to construct point finite covering in collectionwise normal spaces?

Construct a sequence of functions on [0,1] each of which is discontinuous at every point of [0,1] and?

• Construct a sequence of functions on [0,1] each of which is discontinuous at every point of [0,1] and which converges uniformly to a function that is continuous at every point.

• Answer:

  Define f : [0,1] → ℝ by f(x) = 1 for x ∈ [0,1] ∩ ℚ, and f(x) = 0 for x ∈ [0,1] \ ℚ. Now define fn : [0,1] → ℝ for every n ∈ ℕ by fn(x) = f(x)/n. Clearly each fn is everywhere discontinuous, and as a sequence fn converges pointwise to 0 (which is of course continuous). It remains to show that convergence is uniform. Let ε > 0. By the Archimedean property of ℝ there exists a natural number N such that 1/N < ε. Then for all n > N and all x ∈ [0,1] we have |fn(x) - 0| = 1/n < 1/N < ε. → fn uniformly converges to 0.