From f78c5149f93ebc7a123c5c4759f2bf03d0554ebb Mon Sep 17 00:00:00 2001
From: stratisMarkou Chapter 1.ASince \(\epsilon\) can be made arbitrarily small, it follows that \(f\) is Riemann integrable on \([a, b].\)
Exercise 1.A.11
+Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. +Define \(F: [a, b] \to \mathbb{R}\) by
+Prove that \(F\) is continuous on \([a, b].\)
+Let \(t_0 \in [a, b]\) and \(\epsilon > 0.\) +Since \(f\) is Riemann integrable, it is boundedd by some \(C \in \mathbb{R}.\) +For any \(\delta > 0\) and \(x \in [a, b],\) if \(|t - t_0| < \delta,\) then
+Therefore, by picking \(\delta < \epsilon / C\) we have \(|F(t) - F(t_0)| < \epsilon,\) showing that \(F\) is continuous.
+Exercise 1.A.12
+Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. +Prove taht \(|f|\) is Riemann integrable and that
+Let \(\epsilon > 0.\) +Since \(f\) is Riemann integrable, there exists a partition \(P_\epsilon = (x_0, x_1, \dots, x_n)\) of \([a, b]\) such that
+Now since
+we have that
+so \(|f|\) is Riemann integrable. +Since \(U(|f|, P_\epsilon, [a, b]) \geq U(f, P_\epsilon, [a, b]),\) we have that
+The last result in this section concerns a frequently used estimate for the Riemann integral of a function.
-Theorem 58 (Bounds on the Riemann integral)
Suppose \(f: [a, b] \to \mathbb{R}\) is Riemann integrable. diff --git a/prf-prf.html b/prf-prf.html index 5724fcf0..47788b96 100644 --- a/prf-prf.html +++ b/prf-prf.html @@ -423,6 +423,7 @@
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