# 06/15/2019: Real Mathematical Analysis (Charles Chapman Pugh) /home/zack/Dropbox/Library/Charles Chapman Pugh/Real Mathematical Analysis (432)/Real Mathematical Analysis - Charles Chapman Pugh.pdf Last Annotation: 06/15/2019 ## Highlights - 1 Theorem If fn ⇒ f and each fn is continuous at x0 then f is continuous at x0 \. (Charles Chapman Pugh 224) - The uniform limit of continuous functions is continuous\. (Charles Chapman Pugh 224) - Proof (Charles Chapman Pugh 224) - Even if the functions have compact domain of definition, and are uniformly bounded and uniformly continuous, pointwise convergence does not imply uniform convergence\. (Charles Chapman Pugh 225) - The natural way to view uniform convergence is in a function space\. Let Cb = C b \([a, b], R\) denote the set of all bounded functions [a, b] → R\. The elements of Cb are functions f, g, etc\. Each is bounded\. Define the sup norm on Cb as f  = sup{|f \(x\)| : x ∈ [a, b]}\. (Charles Chapman Pugh 225) - 2 Theorem Convergence with respect to the sup metric d is equivalent to uniform convergence\. (Charles Chapman Pugh 227) - 3 Theorem Cb is a complete metric space\. (Charles Chapman Pugh 227) - Let C 0 = C 0 \([a, b], R\) denote the set of continuous functions [a, b] → R (Charles Chapman Pugh 228) - 4 Corollary C 0 is a closed subset of Cb \. It is a complete metric space\. (Charles Chapman Pugh 228) - R\. Each f ∈ C 0 belongs to Cb since a continuous function defined on a compact domain is bounded\. That is, C 0 ⊂ Cb \. (Charles Chapman Pugh 228) - Proof Theorem 1 implies that a limit in Cb of a sequence of functions in C 0 lies in C 0 \. That is, C 0 is closed in Cb \. A closed subset of a complete space is complete\. (Charles Chapman Pugh 228) - 6 Theorem The uniform limit of Riemann integrable functions is Riemann integrable, and the limit of the integrals is the integral of the limit, (Charles Chapman Pugh 229) -  b  b lim fn \(x\) dx = unif lim fn \(x\) dx\. n→∞ a a n→∞ (Charles Chapman Pugh 229) - Proof (Charles Chapman Pugh 229) - 7 Corollary If fn ∈ R and fn ⇒ f then the indefinite integrals converge uniformly, (Charles Chapman Pugh 229)