18 Limits of functions[1HQ]

Definition 404

[2DT]Consider a set \(A\), a function \(f:A\to {\mathbb {R}}\) and a sequence of functions \(f_ n:A\to {\mathbb {R}}\). We will say that \(f_ n\) converges to \(f\) pointwise if

\[ \forall x\in A~ ,~ \lim _{n\to \infty }f_ n(x) = f(x)\quad . \]

We will say that \(f_ n\) converges to \(f\) uniformly if

\[ \forall \varepsilon {\gt}0 \exists N\in {\mathbb {N}}, \forall n \ge N,\forall x\in A~ ,~ |f_ n(x) - f(x)|{\lt}\varepsilon \quad . \]

Further informations on these subjects may be found in Chap. 6 of [ 2 ] , Chap. 11 in [ 4 ] , or Chap. 7 of [ 22 ] .

Definition 405

[1HR] Let \((X_ 1,d_ 1)\) and \((X_ 2,d_ 2)\) be metric spaces. Let \(\mathcal F\) be a family of functions \(f:X_ 1→ X_ 2\), we will say that it is an equicontinuous family if one of these equivalent properties holds.

\(∀ \varepsilon {\gt}0\) \(∃𝛿{\gt}0\) \(∀ f∈\mathcal F\)

\[ ∀ x,y∈ X_ 1,~ d_ 1(x,y)≤ 𝛿⇒ d_ 2(f(x),f(y))≤ \varepsilon \quad . \]
There exists a a fixed monotonically weakly increasing function \(𝜔:[0,∞)→ [0,∞]\), for which \(\lim _{t→ 0+}𝜔(t)=𝜔(0)=0\) (\(𝜔\) is called ”continuity modulus” ¹ ) such that

\begin{equation} ∀ f∈{\mathcal F}, ~ ∀ x,y∈ X_ 1, ~ d_ 2(f(x),f(y))≤ 𝜔\big(d_ 1(x,y)\big)\quad . \label{eq:equicontinua} \end{equation}
406
There exists a fixed continuous function \(𝜔:[0,∞)→ [0,∞]\) with \(𝜔(0)=0\) such that 406 holds.

(The result 6 can be useful to prove equivalence of the last two clauses.)

E406

[1HS] Note:This result is known as ”Dini’s lemma”.
Let \((X,d)\) be a metric space, let \(I⊂ X\) be a compact set, and suppose that \(f,f_ n:I→ℝ\) are continuous and such that \(f_ n(x)↘_ n f(x)\) pointwise (i.e. for every \(x∈ I\) and \(n\) we have \(f(x)≤ f_{n+1}(x) ≤ f_{n}(x)\) and \(\lim _ n f_ n(x) =f(x)\)). Show that \(f_ n→ f\) uniformly.

Hidden solution: [UNACCESSIBLE UUID ’1HT’]

Hidden solution: [UNACCESSIBLE UUID ’1HV’]

In following exercises we will see that, if even one of the hypotheses fails, then there are counterexamples.

E406

[1HW] Find an example of continuous and bounded functions \(f_ n:ℝ→ℝ\) such that \(f_ n(x)↘_ n 0\) pointwise, but not \(f_ n→ 0\) uniformly.

Hidden solution: [UNACCESSIBLE UUID ’1HX’]

E406

[1HY]Find an example of continuous and bounded functions \(f_ n:[0,1]→[0,1]\) such that \(f_ n(x)→_ n 0\) pointwise but not \(f_ n→ 0\) uniformly.

Hidden solution: [UNACCESSIBLE UUID ’1HZ’]

E406

[1J1]Find an example of functions \(f_ n:[0,1]→[0,1]\) continuous, bounded, and such that \(f_ n(x)↘_ n f(x)\) pointwise to \(f:[0,1]→[0,1]\) (i.e. for every \(x\) and \(n\) we have \(0≤ f_{n+1}(x) ≤ f_{n}(x)≤ 1\) and \(\lim _ n f_ n(x) =f(x)\)) but \(f\) is not continuous and the convergence \(f_ n→ f\) is not uniform.

Hidden solution: [UNACCESSIBLE UUID ’1J2’]

E406

[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?

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

Hidden solution: [UNACCESSIBLE UUID ’1J4’]
The convex functions on \(I=[0,1]\).

Hidden solution: [UNACCESSIBLE UUID ’1J5’]
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 405.)

Hidden solution: [UNACCESSIBLE UUID ’1J6’]
Given \(N≥ 0\) fixed, the family of all polynomials of degree less than or equal to \(N\), seen as functions \(f:[0,1]→ℝ\).

Hidden solution: [UNACCESSIBLE UUID ’1J7’]
The regulated functions on \(I=[0,1]\). ²

Hidden solution: [UNACCESSIBLE UUID ’1J9’]
The uniformly continuous and bounded functions on \(I=ℝ\).

Hidden solution: [UNACCESSIBLE UUID ’1JB’]
The Hoelder functions on \(I=[0,1]\), i.e.

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

Hidden solution: [UNACCESSIBLE UUID ’1JC’][UNACCESSIBLE UUID ’1JD’]
The Riemann integrable functions on \(I=[0,1]\).

Hidden solution: [UNACCESSIBLE UUID ’1JF’]

E406

[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.

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. Hidden solution: [UNACCESSIBLE UUID ’1JH’]
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 \([a,b]\).

Hidden solution: [UNACCESSIBLE UUID ’1JJ’]
Let \(f_ n:I→ℝ\) be a family of equicontinuous functions, ³ 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 \([a,b]\).

Hidden solution: [UNACCESSIBLE UUID ’1JK’]
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.

Hidden solution: [UNACCESSIBLE UUID ’1JM’]

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

E406

[1JN] Prerequisites:405, 5 subpoint 6.Difficulty:*.

If \(f_ n,f:I→ℝ\) are uniformly continuous on a set \(I⊂ℝ\), and \(f_ n→_ nf\) uniformly on \(I\), then \(f\) is uniformly continuous, and the family \((f_ n)_ n\) is equicontinuous.

Hidden solution: [UNACCESSIBLE UUID ’1JP’]

E406

[1JQ]Let \(f:ℝ→ℝ\) and let \(g_ t:ℝ→ℝ\) be the translations of \(f\), defined (for \(t∈ℝ\)) by \(g_ t(x)=f(x-t)\). Show that \(g_ t\) tends pointwise to \(f\) for \(t→ 0\), if and only if \(f\) is continuous; and that \(g_ t\) tends uniformly to \(f\) for \(t→ 0\), if and only if \(f\) is uniformly continuous.

Hidden solution: [UNACCESSIBLE UUID ’1JR’]

E406

[1JS] Let \(I⊂ℝ\) be an open set, and let \(\hat x\) be an accumulation point for \(I\) ⁴ , Let \(f_{m}:I→ℝ\) be a sequence of bounded functions that converge uniformly to \(f:I→ℝ\) when \(m→ ∞\). Suppose that, for every \(m\), there exists the limit \(\lim _{x→ \hat x} f_{m}(x)\), then

\[ \lim _{m→ ∞} \lim _{x→\hat x} f_{m}(x)= \lim _{x→\hat x} \lim _{m→ ∞} f_{m}(x) \]

in the sense that if one of the two limits exists, then the other also exists, and they are equal. (The above result also applies to right limits or left limits.)

Show with a simple example that, if the limit is not uniform, then the previous equality does not hold.

Hidden solution: [UNACCESSIBLE UUID ’1JT’] (See also the exercise 5).

E406

[1JV]Let \(I⊂ℝ\) be a compact interval, let \(f_ n,f:I→ℝ\) be continuous. Show that the following two facts are equivalent.

a.: For every \(x∈ X\) and for every sequence \((x_ n)_ n⊂ I\) for which \(x_ n→_ n x\), we have \(\lim _{n→∞} f_ n(x_ n)=f(x)\);
b.: \(f_ n→_ nf\) uniformly on \(I\).

Then find an example where \(I=[0,1)\), the first point holds, but \(f_ n\) does not tend uniformly to \(f\).

Hidden solution: [UNACCESSIBLE UUID ’1JW’]