Sequences of Functions: Pointwise Convergence

Part 2, Chapter 2: Single-Variable Real Analysis

Learning objectives

  • Define pointwise convergence of a sequence {fn}\{f_n\} of functions
  • Compute pointwise limits of common sequences and identify discontinuous limits
  • Demonstrate by example that limit and integral need not commute under pointwise convergence
  • Explain why a stronger notion (uniform convergence) is needed for analysis

Pointwise convergence is the most natural way to take a limit of functions, but it is the WRONG one for analysis. The problem is that the order of "for each xx" and "for each nn" matters enormously: if you fix xx first and then send ntoinftyn\to\infty, the limit f(x)=limfn(x)f(x)=\lim f_n(x)n(x) exists for every xx, but the resulting function ff might be discontinuous even though every fnf_nn is smooth. Worse, you cannot pass limits through integrals or derivatives. The standard example fn(x)=xnf_n(x)=x^nn(x)=xn on [0,1] has continuous fnf_nn but a discontinuous pointwise limit. Pointwise convergence is the warm-up; uniform convergence (next section) is what actually fixes things.

The definition

A sequence of functions fn\{f_n\}n defined on a set SS converges pointwise to a function ff on SS if for every xinSx\in S, the numerical sequence f1(x),f2(x),f_3(x),ldotsf_1(x), f_2(x), f_3(x), \ldots converges to f(x)f(x). Formally: for every xinSx\in S and every epsilon>0\epsilon>0, there exists N(x,epsilon)N(x,\epsilon) such that ∣fn(x)−f(x)∣<epsilon|f_n(x)-f(x)|<\epsilonn(x)−f(x)∣<epsilon for all ngeqNn\geq N. Note crucially that NN may depend on xx, that is the entire weakness of the concept.

The classic example: fn(x)=xnf_n(x)=x^nn(x)=xn on [0,1]

For 0leqx<10\leq x<1, the geometric xnto0x^n\to 0. At x=1x=1, 1n=11^n=1 for every nn. So the pointwise limit is f(x)=0f(x)=0 for xin[0,1)x\in[0,1) and f(1)=1f(1)=1, a discontinuous step function, even though every fnf_nn is a smooth polynomial. The convergence is "slow" near x=1x=1: to get xn<epsilonx^n<\epsilon, you need n>logepsilon/logxn>\log\epsilon/\log x, which blows up as xto1−x\to 1^-. That is precisely why no single NN works for all xx at once.

Why integrals and limits need not commute

Consider fn(x)=nf_n(x)=nn(x)=n on (0,1/n)(0,1/n) and 00 elsewhere on [0,1]. Each fnf_nn has int01fn=1\int_0^1 f_n=11. But pointwise, for every fixed x>0x>0 we have fn(x)=0f_n(x)=0n(x)=0 for all sufficiently large nn, so the pointwise limit is fequiv0f\equiv 0. Therefore limnintfn=1neq0=intlimnfn\lim_n \int f_n=1\neq 0=\int\lim_n f_nn=1neq0=intlimnfn. The mass "escapes to a point", a phenomenon unique to limits of unbounded sequences. This is the failure mode pointwise convergence cannot detect.

Use the grapher to plot fn(x)=xnf_n(x)=x^nn(x)=xn on [0,1] for n=1,2,5,10,50n=1, 2, 5, 10, 50. Watch how each curve hugs the xx-axis for small xx and then sharply rises to 1 near x=1x=1. As nn grows, the transition becomes a near-vertical wall just left of 1. The pointwise limit is the step function (0 for x<1x<1, 1 at x=1x=1), but no continuous curve "looks like" the step function for any finite nn, that is the visual signature of failed uniform convergence.

Where this shows up
  • Numerical integration & convergence theory: the Lebesgue Dominated Convergence Theorem and the Monotone Convergence Theorem give exactly the conditions under which pointwise convergence of integrands implies convergence of integrals. Numerical analysts rely on these results when they pass to the limit in Monte Carlo, finite-element, or spectral methods.
  • Approximation theory: a sequence of approximation operators (Bernstein polynomials, Fourier partial sums, splines) is typically constructed to converge pointwise to a target function; whether the convergence is uniform determines whether the approximation is acceptable for engineering applications.
  • Probability & convergence of distributions: a sequence of random variables converges "in distribution" iff their CDFs converge pointwise at every continuity point. The Central Limit Theorem is fundamentally a pointwise-CDF-convergence statement.

Pause and think: Why does pointwise convergence "lose" continuity of the limit even when each fnf_nn is continuous? Try to articulate, in your own words, what extra control we would need on fn\{f_n\}n to guarantee the limit is continuous.

Try it

  • Predict first: find the pointwise limit of fn(x)=x/nf_n(x)=x/nn(x)=x/n on mathbbR\mathbb{R}. Is the limit continuous? Is the convergence uniform? (Spoiler: limit is 0, yes continuous, no uniform on all of mathbbR\mathbb{R}.)
  • Compute the pointwise limit of fn(x)=1/(1+x2n)f_n(x)=1/(1+x^{2n})n(x)=1/(1+x2n) on [0,infty)[0,\infty). (Hint: split into x<1x<1, x=1x=1, x>1x>1.) Is the limit continuous?
  • Construct a sequence of continuous fnf_nn on [0,1] with int01fn=1\int_0^1 f_n=11 but fnto0f_n\to 0nto0 pointwise. (Tent functions of height nn and base 2/n2/n work.)
  • True or false: if fntoff_n\to fntof pointwise on [a,b] and each fnf_nn is bounded, then ff is bounded. Justify or give a counterexample.

    A trap to watch for

    Pointwise convergence is so weak that most "nice" theorems about limits and analysis ARE FALSE under it. Specifically: pointwise limit of continuous can be discontinuous; pointwise limit of integrable can be non-integrable; limintneqintlim\lim\int\neq\int\lim in general; limfn′neq(limfn)′\lim f_n'\neq(\lim f_n)'n)′ in general. Whenever you want to swap a limit with another operation, you need a stronger hypothesis, usually uniform convergence (§2.7) or a dominated/monotone hypothesis (Lebesgue integration).

    What you now know

    You can compute pointwise limits, construct discontinuous-limit examples, build counterexamples to interchange-of-limits, and articulate why pointwise convergence alone is insufficient. Section §2.7 introduces uniform convergence, which is exactly the upgrade needed: it preserves continuity, allows interchange with integration, and (with a derivative condition) allows interchange with differentiation.

    Mark section complete →

    References

    • Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 2.
    • Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 7.
    • Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer, ch. 6.
    • Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 9.
    • Tao, T. (2016). Analysis I (3rd ed.). Hindustan Book Agency, ch. 14.

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