Problem Set 14

  1. Medium
    Using any equivalent definition of integration (see Question 4), show that integration is a linear operator; that, show that if \(f_1\) and \(f_1\) are integrable on \([a,b]\), then \(c_1 f_1+ c_2f_2\) is integrable on \([a,b]\) and \[ \int_a^b \left[ c_1f_1(x) + c_2f_2(x) \right]dx = c_1 \int_a^b f_1(x) dx + c_2 \int_a^b f_2(x) dx. \]

  2. Medium
    Show that the set \(S = \left\{\frac{1}{n}\right\}_{n=1}^\infty\subseteq \mathbb R\) has Jordan measure zero.

  3. Medium
    More generally, show that if \((a_n)_{n=1}^\infty\) is any convergent sequence, then \((a_n)\) has Jordan measure zero.

  4. Hard
    Prove that the following statements are equivalent:
    1. The bounded function \(f: [a,b] \to \mathbb R\) satisfies the following: there is a real number \(I\) such that, for all \(\epsilon >0\) there exists a \(\delta>0\) having the property that whenever \(P \in \mathcal P_{[a,b]}\) is a partition of \([a,b]\) with \(\ell(P) < \delta\), then \[ |S(f,P) - I| < \epsilon. \]
    2. If \(f:[a,b] \to \mathbb R\) is a bounded function, then \[ \sup_{P\in\mathcal P_{[a,b]}} L_f(P) = \inf_{P\in\mathcal P_{[a,b]}} U_f(P). \]
    3. The bounded function \(f:[a,b] \to \mathbb R\) satisfies the following property: For every \(\epsilon >0\) there exists a partition \(P\in\mathcal P_{[a,b]}\) such that \[ U_f(P) - L_f(P) < \epsilon. \]
    4. The bounded function \(f:[a,b] \to \mathbb R\) satisfies the property that for every \(\epsilon >0\), there exists a \(\delta >0\) such that whenever \(P,Q \in \mathcal P_{[a,b]}\) are two partitions satisfying \(\ell(P)<\delta, \ell(Q)<\delta\) then \[ |S(f,P) - S(f,Q)| < \epsilon. \]

  5. Hard
    Show that every Riemann integrable function \(f:[a,b] \to \mathbb R\) is bounded; that is, there exists some \(M>0\) such that \(|f(x)|\leq M\) for all \(x \in [a,b]\). (NOTE: feel free to SKIP this question, as some of the lectures defined Riemann integrable over closed rectangles only for bounded functions to begin with)

  6. Hard
    Show that if \(f:[a,b] \to \mathbb R\) is Riemann integrable, then \(F(x) = \int_a^x f(s) ds\) is uniformly continuous on \([a,b]\). [Hint: Use the fact that integrable functions are bounded, together with a couple of properties for integrals, one involving an absolute value]

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