EDB — 1J3

[1J3]

Let \(I⊂ℝ\) be an interval. Which of these classes \(\mathcal F\) of functions \(f:I→ℝ\) are closed for uniform convergence? Which are closed for pointwise convergence?

1. The continuous and monotonic (weakly) increasing functions on \(I=[0,1]\).

   Solution 1

   [1J4]↺↻

2. The convex functions on \(I=[0,1]\).

   Solution 2

   [1J5]↺↻

3. Given \(𝜔:[0,∞)→ [0,∞)\) a fixed continuous function with \(𝜔(0)=0\) (which is called ”continuity modulus”), and

   \[ {\mathcal F}=\{ f:[0,1]→ℝ ~ :~ ∀ x,y, |f(x)-f(y)|≤ 𝜔(|x-y|)\} \]

   (this is called a family of equicontinuous functions, as explained in the definition [1HR]↺↻.)

   Solution 3

   [1J6]↺↻

4. Given \(N≥ 0\) fixed, the family of all polynomials of degree less than or equal to \(N\), seen as functions \(f:[0,1]→ℝ\).

   Solution 4

   [1J7]↺↻

5. The regulated functions on \(I=[0,1]\). ¹

   Solution 5

   [1J9]↺↻

6. The uniformly continuous and bounded functions on \(I=ℝ\).

   Solution 6

   [1JB]↺↻

7. The Hoelder functions on \(I=[0,1]\), i.e.

   \[ \Big\{ f:[0,1]→ℝ ~ \Big|~ ∃ b{\gt}0,∃𝛼∈(0,1]~ ~ ∀ x,y∈[0,1], |f(x)-f(y)|≤ b |x-y|^𝛼\Big\} \]
   Solution 7

   [1JC]↺↻

8. The Riemann integrable functions on \(I=[0,1]\).

   Solution 8

   [1JF]↺↻