EDB — 1JG

[1JG] We wonder if the previous classes \(\mathcal F\) enjoy a ”rigidity property”, that is, if from a more "weak" convergence in the class follows a more "strong" convergence. Prove the following propositions.

1. Let \(f_ n,f:I→ℝ\) be continuous and monotonic (weakly) increasing functions, defined over a closed and bounded interval \(I=[a,b]\). Suppose there is a dense set \(J\) in \(I\) with \(a,b∈ J\), such that \(∀ x∈ J, f_ n(x)→_ n f(x) \), then \(f_ n→_ nf\) uniformly.

   Solution 1

   [1JH]↺↻

2. Let \(A⊆ ℝ\) be open interval. Let \(f_ n,f:A→ℝ\) be convex functions on \(A\). If there is a set \(J\) dense in \(A\) such that \(∀ x∈ J, f_ n(x)→_ n f(x) \), then, for every \([a,b]⊂ A\), we have that \(f_ n→_ n f\) uniformly on \([a,b]\).

   Solution 2

   [1JJ]↺↻

3. Let \(f_ n:I→ℝ\) be a family of equicontinuous functions, ¹ defined on a closed and bounded interval \(I=[a,b]\), and let \(𝜔\) be their modulus of continuity. If there is a set \(J\) dense in \([a,b]\) such that \(∀ x∈ J, f_ n(x)→_ n f(x) \), then, \(f\) extends from \(J\) to \(I\) so that it is continuous (with modulus \(𝜔\)), and \(f_ n→_ nf\) uniformly on \([a,b]\).

   Solution 3

   [1JK]↺↻

4. Let \(f_ n,f:I→ℝ\) be polynomials of degree less than or equal to \(N\), seen as functions defined on an interval \(I=[a,b]\) closed and bounded; fix \(N+1\) distinct points \(a≤ x_ 0{\lt}x_ 1{\lt}x_ 2{\lt}\ldots {\lt}x_ N≤ b\); assume that, for each \(x_ i\), \(f_ n(x_ i)→_ n f(x_ i)\): then \(f_ n\) converge to \(f\) uniformly, and so do each of their derivatives \(D^ kf_ n→_ n D^ kf\) uniformly.

   Solution 4

   [1JM]↺↻

Also look for counterexamples for similar propositions, when applied to the other classes of functions seen in the previous exercise.