Uniform convergence

When can we swap a limit with continuity, integration or differentiation?

1. Pointwise convergence (Def. 1.1) is too weak (Example 1.2).
2. Uniform convergence (Def. 1.3) and its sup-norm test (Prop. 1.4) and Cauchy criterion (Prop. 1.5).
3. Uniform limits of continuous functions are continuous (Thm 1.6); limits commute with integrals (Thm 1.7) and, with care, derivatives (Thm 1.8).
4. Series of functions and the Weierstrass M-test (Thm 1.9).

$f_n\to f$ pointwise on $S$ if $f_n(x)\to f(x)$ for each $x\in S$: for every $x$ and $\varepsilon > 0$ there is $N$ (depending on $x$ and $\varepsilon$) with $|f_n(x) - f(x)| < \varepsilon$ for $n\ge N$.

• Continuity: $x^n$ on $[0, 1]$ has a discontinuous limit.
• Integrals: $f_n = n^2x(1 - x^2)^n$ on $[0, 1]$ tends to $0$ pointwise, yet $\int_0^1f_n = \frac{n^2}{2(n + 1)}\to\infty$.
• Derivatives: $f_n = \frac{\sin(n^2x)}{n}\to0$ (even uniformly), but $f_n'(0) = n\to\infty$.

$f_n\to f$ uniformly on $S$ if for every $\varepsilon > 0$ there is $N$ (independent of $x$) with $|f_n(x) - f(x)| < \varepsilon$ for all $n\ge N$ and all $x\in S$.

$f_n\to f$ uniformly on $S$ iff $\|f_n - f\|_\infty = \sup_{x\in S}|f_n(x) - f(x)|\to0$. To disprove uniformity it suffices to find $x_n\in S$ with $|f_n(x_n) - f(x_n)|\not\to0$.

$(f_n)$ converges uniformly on $S$ (to some $f$) iff for every $\varepsilon > 0$ there is $N$ with $\|f_n - f_m\|_\infty < \varepsilon$ for all $m, n\ge N$.

If each $f_n$ is continuous at $a\in S$ and $f_n\to f$ uniformly on $S$, then $f$ is continuous at $a$.

If $f_n$ are Riemann integrable on $[a, b]$ and $f_n\to f$ uniformly, then $f$ is integrable and $\int_a^bf_n\to\int_a^bf$.

Let $f_n$ be continuously differentiable on $[a, b]$, with $f_n'\to g$ uniformly and $f_n(c)$ convergent for some $c\in[a, b]$. Then $f_n$ converges uniformly to a differentiable $f$ with $f' = g$.

If $|g_k(x)|\le M_k$ for all $x\in S$ and $\sum M_k < \infty$, then $\sum g_k$ converges uniformly (and absolutely) on $S$. If the $g_k$ are continuous, so is the sum.

• Find the pointwise limit first.
• Compute $\sup|f_n - f|$ (calculus: differentiate in $x$).
• Disprove: a bad sequence $x_n$, or a discontinuous limit of continuous functions.
• Series: the M-test; if it fails, check the tail directly.
• Uniformity depends on the set: shrinking the domain often helps.