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 $f_n,f:I\to ℝ$ 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 $a,b\in J$, such that $\forall x\in J,f_n(x){\to }_{n}f(x)$, then $f_n{\to }_{n}f$ uniformly.

   Solution 1

   [1JH]↺↻

2. Let $A\subseteq ℝ$ be open interval. Let $f_n,f:A\to ℝ$ be convex functions on $A$. If there is a set $J$ dense in $A$ such that $\forall x\in J,f_n(x){\to }_{n}f(x)$, then, for every $[a,b]\subset A$, we have that $f_n{\to }_{n}f$ uniformly on $[a,b]$.

   Solution 2

   [1JJ]↺↻

3. Let $f_n:I\to ℝ$ 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 $\forall x\in J,f_n(x){\to }_{n}f(x)$, then, $f$ extends from $J$ to $I$ so that it is continuous (with modulus $𝜔$), and $f_n{\to }_{n}f$ uniformly on $[a,b]$.

   Solution 3

   [1JK]↺↻

4. Let $f_n,f:I\to ℝ$ 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 $a\le x_0<x_1<x_2<\dots <x_N\le b$; assume that, for each $x_i$, $f_n(x_i){\to }_{n}f(x_i)$: then $f_n$ converge to $f$ uniformly, and so do each of their derivatives $D^k{f}_n{\to }_n{D}^{k}f$ uniformly.

   Solution 4

   [1JM]↺↻

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