Real Analysis 24 | Pointwise Convergence [dark version]

Q: Why does the example F_n(x)=1/(n(x+1)) converge pointwise to 0 on [0,1]?

Fix any x̃ in [0,1]. Then F_n(x̃)=1/(n(x̃+1)) is a constant multiple of 1/n, so it tends to 0 as n→∞. Since this holds for every x̃, the pointwise limit function is f(x)=0 for all x in [0,1].

Q: In the piecewise parabola example, how can the peak height grow while the pointwise limit is still zero?

For x in [0,1/n], F_n(x)=n^2 x(1−nx), a parabola with zeros at x=0 and x=1/n. Its maximum occurs at x=1/(2n), where F_n(1/(2n))=n/4, so the peak height increases with n. But for any fixed x>0, once n>1/x, that x lies in [1/n,1], where F_n(x)=0. So for each fixed x, the sequence becomes 0 eventually, giving pointwise limit 0 everywhere.

TL;DR

Pointwise convergence requires that for every fixed x in the domain, the numerical sequence F_n(x) converges to f(x).

Briefing Cornell Notes

Briefing

Pointwise convergence can look “well-behaved” on every fixed input, yet still produce a limit function with surprising features—so it’s not strong enough for many analysis problems. The key takeaway is that a sequence of functions can converge to a limit at each point x (vertical slices converge), even while the functions develop increasingly sharp behavior or even change qualitative shape in the limit. That mismatch is exactly why uniform convergence is introduced as the stronger, more reliable notion.

The discussion starts by recalling the definition of pointwise convergence for a sequence of functions {F_n} on an interval I. For every fixed x̃ in I, the numerical sequence F_n(x̃) must converge. Equivalently, for every ε>0 and every x̃, there exists an index N (which may depend on x̃) such that for all n≥N, |F_n(x̃)−f(x̃)|<ε. This quantifier structure matters: the “eventually” index can vary with the point x̃.

A first example uses F_n(x)=1/(n(x+1)) on I=[0,1]. Fixing x̃ turns the problem into a number sequence 1/(n(x̃+1)), which converges to 0 as n→∞. Since this happens for every x̃, the pointwise limit is the constant zero function. Here, the graphs flatten out as n grows, matching the limit.

The next example is designed to show how pointwise convergence can hide growing peaks. On I=[0,1], define F_n piecewise: for x in [0,1/n], F_n(x)=n^2 x(1−nx); for x in [1/n,1], F_n(x)=0. Each F_n is zero on the right side and forms a parabola on the left interval whose width shrinks like 1/n. The maximum occurs at x=1/(2n), giving F_n(1/(2n))=n/4, so the peak height increases with n even though the support shrinks.

Despite the peak growing without bound, the pointwise limit still exists and equals 0 everywhere. If x=0, every F_n(0)=0. If x>0, then for sufficiently large n (specifically n>1/x), the point x lies in the region where the function is identically zero, so F_n(x)=0 eventually. Thus, for each fixed x, the sequence is eventually constant at 0, yielding pointwise convergence to the zero function.

This is the central “strange result”: pointwise convergence can coexist with functions that get taller and more concentrated. The transcript then gestures at an even more qualitative issue: a sequence of continuous functions can converge pointwise to a limit function with a jump discontinuity. That possibility—where the limit can develop behavior not present in any F_n—motivates uniform convergence as the next step, a stronger condition that prevents such pathological outcomes. The formal definition and examples of uniform convergence are deferred to the next video.

Cornell Notes

Pointwise convergence means that for every fixed x in the domain, the numerical sequence F_n(x) converges to a limit f(x). The quantifiers allow the “eventually” index N to depend on x, which can let sharp features move around as n grows. In the example F_n(x)=n^2 x(1−nx) on [0,1/n] and 0 on [1/n,1], the peak height grows like n/4 while the peak width shrinks like 1/n. Even so, for any fixed x>0, eventually x lies outside the shrinking interval, so F_n(x)=0 for all large n. The limit is therefore the zero function everywhere, illustrating that pointwise convergence can miss growing or qualitative changes that only appear in the limit.

What is the precise definition of pointwise convergence for a sequence of functions?

For functions F_n on a domain I, pointwise convergence to f means: for every x̃ in I and every ε>0, there exists an index N (which may depend on x̃) such that for all n≥N, |F_n(x̃)−f(x̃)|<ε. Equivalently, each “vertical slice” x̃↦F_n(x̃) is a convergent numerical sequence.

Why does the example F_n(x)=1/(n(x+1)) converge pointwise to 0 on [0,1]?

Fix any x̃ in [0,1]. Then F_n(x̃)=1/(n(x̃+1)) is a constant multiple of 1/n, so it tends to 0 as n→∞. Since this holds for every x̃, the pointwise limit function is f(x)=0 for all x in [0,1].

In the piecewise parabola example, how can the peak height grow while the pointwise limit is still zero?

For x in [0,1/n], F_n(x)=n^2 x(1−nx), a parabola with zeros at x=0 and x=1/n. Its maximum occurs at x=1/(2n), where F_n(1/(2n))=n/4, so the peak height increases with n. But for any fixed x>0, once n>1/x, that x lies in [1/n,1], where F_n(x)=0. So for each fixed x, the sequence becomes 0 eventually, giving pointwise limit 0 everywhere.

What role does the shrinking interval [0,1/n] play in the convergence?

The region where F_n is nonzero shrinks to the single point x=0. That means points x>0 eventually fall into the region where the function is identically zero. Pointwise convergence only checks behavior at each fixed x, so it can ignore what happens in a neighborhood that keeps moving/shrinking with n.

Why does the transcript claim pointwise convergence can be “not strong enough”?

Because pointwise convergence can allow limit behavior that doesn’t reflect uniform control across the domain. The transcript highlights that even if every F_n is continuous (no jumps), the pointwise limit can develop a jump discontinuity. That motivates uniform convergence, which restricts how quickly and where deviations can persist across all x simultaneously.

Review Questions

In the definition of pointwise convergence, which quantity is allowed to depend on the point x̃: N or ε?
For the piecewise function on [0,1], what inequality on n ensures that a fixed x>0 lies in the region where F_n(x)=0?
How does the example with peak height n/4 illustrate the difference between pointwise convergence and uniform control?

Key Points

1
Pointwise convergence requires that for every fixed x in the domain, the numerical sequence F_n(x) converges to f(x).
2
The index N in the ε-definition of pointwise convergence may depend on x, which weakens the control over the whole domain.
3
For F_n(x)=1/(n(x+1)) on [0,1], the pointwise limit is the zero function because 1/n→0 for every fixed x.
4
A sequence can have peaks that grow in height (like n/4) while still converging pointwise to 0 if the nonzero region shrinks to a point.
5
In the piecewise parabola example, any x>0 eventually lies outside [0,1/n], forcing F_n(x)=0 for all sufficiently large n.
6
Pointwise convergence can permit qualitative changes in the limit, such as a jump discontinuity arising from continuous functions, motivating uniform convergence as the next stronger concept.

Highlights

Pointwise convergence checks each x separately, so the “eventually” time can vary across the domain.

In the parabola example, the maximum value is n/4 at x=1/(2n), yet the pointwise limit is still 0 everywhere.

A shrinking support [0,1/n] can make F_n(x)=0 eventually for every fixed x>0, even while the graphs develop taller peaks.

Continuous functions can converge pointwise to a discontinuous limit, showing why a stronger notion is needed.

Topics

Pointwise Convergence
Uniform Convergence
Sequences of Functions
ε-Definition
Limit Functions