連續性在一致極限下存活
回報來了。一致極限定理說：若每個 f_n 在點 a 處連續，且 f_n -> f 一致，則 f 在 a 處連續。證明是著名的 epsilon/3 論證——三個誤差，各壓到 epsilon/3 以下，用三角不等式黏合。一致性恰好使其中兩個誤差與點無關。
Uniform limit theorem (continuity at a).
Given: f_n -> f uniformly on E, each f_n continuous at a in E.
Claim: |f(x) - f(a)| < epsilon for x near a.
Fix epsilon > 0.
(1) UNIFORMITY: choose N with ||f_N - f|| < epsilon/3, so for ALL x
|f(x) - f_N(x)| < epsilon/3.
(2) CONTINUITY of f_N at a: choose delta > 0 with
|f_N(x) - f_N(a)| < epsilon/3 whenever |x - a| < delta.
Now for |x - a| < delta, split with the triangle inequality:
|f(x) - f(a)|
<= |f(x) - f_N(x)| + |f_N(x) - f_N(a)| + |f_N(a) - f(a)|
< epsilon/3 + epsilon/3 + epsilon/3
= epsilon.
Hence f is continuous at a. QED
(Errors 1 and 3 used the SAME N for all x — that is uniformity.)逐項積分
在有界閉區間 [a,b] 上，若Riemann 可積的 f_n 一致收斂到 f，則 f 可積，且積分與極限可交換。這就是逐項積分。證明只需一行放縮：兩積分之差至多為區間長度乘上上確界範數誤差。
Term-by-term integration on [a,b], f_n -> f uniformly. | integral_a^b f_n - integral_a^b f | = | integral_a^b (f_n - f) | <= integral_a^b |f_n - f| <= integral_a^b ||f_n - f|| (constant bound) = (b - a) * ||f_n - f|| -> 0. So lim_n integral_a^b f_n = integral_a^b lim_n f_n = integral_a^b f. WARNING (needs uniform!): the spike g_n(x) = n x (1-x)^n on [0,1] from Guide 2 converges pointwise to 0, yet integral_0^1 g_n -> a nonzero constant, NOT 0. Without uniformity the swap is false.
求導更難伺候
f_n 的一致收斂不給出 f_n' -> f'。導數感受斜率，振幅極小的抖動也可有巨大斜率。正確的逐項求導定理把假設翻轉：假定導數 f_n' 一致收斂，且 (f_n) 在哪怕一點收斂。則 f_n -> f 一致，並且關鍵地 f' = lim f_n'。
Counterexample: amplitude small, slope not. f_n(x) = sin(n x) / sqrt(n) on R. ||f_n|| = 1/sqrt(n) -> 0, so f_n -> 0 UNIFORMLY. But f_n'(x) = sqrt(n) cos(n x), and f_n'(0) = sqrt(n) -> infinity. So f_n' does NOT converge to (0)' = 0; it diverges. Moral: control the DERIVATIVES uniformly, not the functions, to differentiate term by term. Even scarier (Guide 5 preview): a uniform limit (a Weierstrass series sum b^k cos(a^k pi x), 0<b<1, ab>1) can be continuous yet NOWHERE differentiable.