Commuting Limiting Operations

\begin{align*} \lim_{m\to \infty}\lim_{n\to\infty} a_{mn} \neq \lim_{n\to \infty}\lim_{m\to\infty} a_{mn} .\end{align*}

For example, \begin{align*} \lim_{m\to \infty}\lim_{n\to\infty} {n \over n + m} &= 1 \neq 0 = \lim_{n\to \infty}\lim_{m\to\infty} {n \over n + m} .\end{align*}

\begin{align*} \lim_{n\to \infty}\sup_{x\in X} {\left\lvert {f_n(x) } \right\rvert} \neq \sup_{x\in X} {\left\lvert {\lim_{n\to\infty} f_n(x) } \right\rvert} .\end{align*}

\begin{align*} \lim_{k\to \infty} \lim_{n\to\infty} f_n(x_k) \neq \lim_{n\to \infty} \lim_{k\to\infty} f_n(x_k) .\end{align*} For example, take \(f_n(x) \coloneqq x^n\), then for \(\left\{{x_k}\right\}\to 1\), \begin{align*} \lim_{k\to\infty}\lim_{n\to\infty} (x_k)^n &= \lim_{k\to \infty } \chi_{x=1}(x_k) = 0 \\ \lim_{n\to\infty } \lim_{k\to\infty } (x_k)^n &= \lim_{n\to\infty} (1)^n = 1 .\end{align*}

\begin{align*} \lim_{n\to \infty} {\frac{\partial }{\partial x}\,} f_n \neq {\frac{\partial }{\partial n}\,} \qty{\lim_{n\to \infty} f_n} .\end{align*} For example, \begin{align*} f_n(x) \coloneqq\sqrt{x^2 + {1\over n}} \overset{n\to\infty}\longrightarrow f(x) \coloneqq{\left\lvert {x} \right\rvert} ,\end{align*} and this convergence is even uniform.

Another example: \begin{align*} f_n(x) \coloneqq{x\over 1 + nx^2} .\end{align*} Then by Calculus, \(f_n(x) \leq 1/2\sqrt{n} \coloneqq M_n\) and \(f_n\to 0\) uniformly, so \(f' = 0\). But \begin{align*} f_n'(x) = {1-nx^2 \over\qty{1 + nx^2}^2} ,\end{align*} and \(f_n'(0) \to 1\).

Note that uniform convergence of \(f_n\) and \(f_n'\) is sufficient to guarantee that \(f\) is differentiable. Even worse: every continuous function is a uniform limit of polynomials by the Weierstrass approximation theorem.

\begin{align*} \lim_{n\to \infty} \int_a^b f_n(x) \,dx \neq \int_a^b \lim_{n\to \infty} \qty{ f_n(x) } \,dx .\end{align*}

  • Differentiability \(\implies\) continuity but not the converse: \(f(x) = {\left\lvert {x} \right\rvert}\).

    • The Weierstrass function is continuous but nowhere differentiable.
  • \(f\) continuous does not imply \(f'\) is continuous: \(f(x) = x^2 \sin(x)\).

  • Limit of derivatives may not equal derivative of limit: \begin{align*} f(x) = \frac{\sin(nx)}{n^c} \text{ where } 0 < c < 1. \end{align*}

    • Also shows that a sum of differentiable functions may not be differentiable.
  • Limit of integrals may not equal integral of limit: \begin{align*} \sum \indic{x = q_n \in {\mathbb{Q}}} .\end{align*}

  • A sequence of continuous functions converging to a discontinuous function: \begin{align*} f(x) = x^n \text{ on } [0, 1] .\end{align*}