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\[B_{\epsilon}=\{x_{1} Assume for now \(a \[\mathcal{P}_{0}=\left\{a,x_{i}-\frac{1}{2m},x_{i}+\frac{1}{2m},b\right\}\;.\] We have \[0\leq\sup\left\{f(x):\;x_{i}-\frac{1}{2m}\leq x\leq x_{i}+\frac{1}{2m}\right\}\leq 1\] because \(0\leq f(x)\leq 1\) for all \(x\in[a,b]\). On any other interval \(I\) associated with the partition, we have \[0\leq\sup\left\{f(x):\;x\in I\right\}\leq\frac{\epsilon}{2(b-a)}\] because \(I\cap B_{\epsilon}\) is empty. Hence \[0\leq U(\mathcal{P}_{0},f)\leq n\frac{1}{m}+\frac{\epsilon}{2(b-a)}(b-a)= \frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon\;.\] If \(x_{1}=a\), then we consider only the interval \([a,a+1/m]\), and if \(x_{n}=b\), then we consider only the interval \([b-1/m,b]\). The proof is carried similarly to get \(U(\mathcal{P}_{0},f)\leq\epsilon\). Clearly this will imply \(\inf U(\mathcal{P},f)=0\), where the infimum is taken over all partitions of \([a,b]\). Hence \[L(f)=U(f)=0\;,\] which implies that \(f(x)\) is Riemann integrable and \(\int_{a}^{b}f(x)dx=0\). **Theorem 53** (Lebesgue’s Theorem).: _Let \(f\) be a bounded function defined on \([a,b]\). Then \(f\) is Riemann integrable if and only if the set of points where \(f\) is not continuous has measure zero._ The proof of Lebesgue’s theorem depends on making several observations on the set of oscillations of \(f\) and finding a partition \(P\) so that \(U(f)-L(f)<\epsilon\). For the details of the proof, consult [52], p. 473. Using Lebesgue's theorem, we can decide the Riemann integrability of certain functions quite easily. For example, if we have \[f(x)=\left\{\begin{array}{ll}\sin\left(\frac{1}{x}\right)&\mbox{ if }x\neq 0\\ 0&\mbox{ if }x=0,\end{array}\right.\] then we can claim \(f\) is Riemann integrable on \([-1,1]\). Note that the length of \([-1,1]\neq 0\) and \(f\) has one point of discontinuity at \(x=0\); thus the set of discontinuities has measure zero. Additionally \(f\) is bounded since \(|f(x)|\leq 1\); by the above theorem it follows that it is Riemann integrable. The set of discontinuities for the function in Example 54 is \(\mathbb{Q}\). Since \(\mathbb{Q}\) is countable, it has measure zero, so we again conclude that \(f\) is Riemann integrable. **Remark 30**.: For the Riemann integral we divide the area under the graph of \(f\) into vertical rectangles to find upper and lower sums. How about dividing this area into horizontal rectangles? This leads to a more complicated mathematics. Suppose \(f:[a,b]\to\mathbb{R}\) is a nonnegative bounded function defined on \([a,b]\). Let \(R=\{y_{0} Suppose we want to find out \[\sum_{k=1}^{n}(y_{k+1}-y_{k})\text{ (length of the interval}\,\{x:f(x)\geq y_{k}\}).\] However, the set \(\{x:f(x)\geq y_{k}\}\) might be a complicated set, and one asks how can one find its length? To answer this question one has to develop the idea of nonzero measure (or length) of a set. Among many notions of integral, the most prominent one is the Lebesgue integral. To develop these ideas precisely requires a text in itself (see [50]). We return to the idea of Lebesgue integral in Chapter 2, Section 2.7. ### Exercises 1. Let \(f(x)=1-x^{2}\). Compute \(U(f,P)\), \(L(f,P)\) and \(\int_{0}^{1}f(x)\,dx\), where \[P=\left\{0,\frac{2}{5},\frac{1}{2},\frac{3}{5},1\right\}.\] Figure 1.49: Dividing the area under \(f\) into horizontal rectangles [MISSING_PAGE_EMPTY:459] * What is \(F(x)=\int_{0}^{x}f(t)\,dt\) on \([0,2]\)? 10. Let \(f:[a,b]\to\mathbb{R}\) be a Riemann integrable function. Let \(g:[a,b]\to\mathbb{R}\) be a function such that \(\{x\in[a,b];f(x)\neq g(x)\}\) is finite. Show that \(g(x)\) is Riemann integrable and \[\int_{a}^{b}f(x)\,dx=\int_{a}^{b}g(x)\,dx\;.\] Does the conclusion still hold when \(\{x\in[a,b];f(x)\neq g(x)\}\) is countable? 11. Suppose that \(f\) is Lipschitz with constant \(L\) on \([a,b]\). Prove that \[\left|\int_{0}^{1}f(x)\,dx-\frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n})\right|\leq \frac{L}{n}.\] 12. We say \(f\) is Riemann integrable on \([a,\infty)\) if \(\lim_{b\to\infty}\int_{a}^{b}f(x)\,dx\) exists. Use \(\int_{a}^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_{a}^{b}f(x)\,dx\), to show that \(\int_{0}^{\infty}\frac{\sin x}{x}\,dx\) exists. ## Chapter 2 Real Analysis The following theorem can be found in the work of Mr. Cauchy: “If the various terms of the series \(u_{1}+u_{2}+\cdots\) are continuous functions, then the sum \(s\) of the series is also a continuous function of \(x\).” But it seems to me that this theorem admits exceptions. For example the series \[\sin x-\frac{1}{2}\sin 2x+\frac{1}{3}\sin 3x…\] is discontinuous at each value \((2m+1)\pi\) of \(x\),… _Abel, 1826, Oeuvres, vol. 1, p. 224-225_ ### 2.1 Metric, Normed, and Inner Product Spaces Think about all of our usual operations, starting from limits, derivatives, integrals, sums, then all operations on sets and functions, vectors, matrices, etc. Instead of dealing with individual operations, M. Frechet, together with other mathematicians of the twentieth century, developed abstract spaces, where these operations were considered as functions defined on entire collections of abstract objects. As early as 1906, in his thesis, “Sur quelques points du calcul fictionnel,” Frechet introduced a notion of distance defined on abstract sets of points (Figure 2.1). He considered the collection \(C[0,1]\) of continuous functions \(f:[0,1]\to\mathbb{R}\) where one measures the distance between functions \(d(f,g)\) by taking the maximum vertical distance between their graphs, that is: \[d(f,g)=\max_{0\leq x\leq 1}|f(x)-g(x)|.\]In 1928, Frechet also wrote a monograph on his research of abstract spaces called “Les Espaces Abstraits.” This monograph contains most of the examples of abstract metric spaces. The other two spaces of interest are normed and inner product spaces. In 1922, S. Banach wrote down the axioms for normed spaces and axioms of inner product spaces, which were later presented by John von Neumann. In the following, we first define a _”metric space,” “inner product space,”_ and _”normed space.”_ We will see that every inner product space is a normed space and every normed space is a metric space. All of these abstract spaces are generally called _”topological spaces.”_ ### Metric Spaces In this section we will introduce the notion of a metric space. The word _metric_ is simply a synonym for _distance_. **Definition 39** (Metric).: Given a nonempty set \(M\), a function \[d:M\times M\to\mathbb{R}\] satisfying the following properties is called a _metric_ on \(M\). * \(0\leq d(x,y)<\infty\) for all pairs \(x,y\in M\).
* \(d(x,y)=0\Leftrightarrow x=y\).
* \(d(x,y)=d(y,x)\) for all pairs \(x,y\in M\).
* \(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z\in M\) (Triangle Inequality). We can use this function to define the concept of a metric space. **Definition 40** (Metric Space).: The couple \((M,d)\) consisting of a set together with a metric \(d\) defined on \(M\) is called a _metric space_. There are many different kinds of metric spaces, some of which play very significant roles in geometry and analysis. We will now look at some common examples of metric spaces. Our first example is a rather trivial metric, but it shows every set \(M\) admits at least one metric. Figure 2.1: Max vertical distance **Example 55**.: Let \(M\) be an arbitrary nonempty set. Define \(d\) by
2. Suppose \(f\) is continuous on \(\mathbb{R}\). Explain why the functions defined by \(f^{3}(x)\), \(\cos(f(x))\), or \(f(\cos x)\) are all integrable over every interval \([a,b]\).
* Is \(F(x)\) continuous?
* Is \(F^{\prime}(x)=f(x)\)?
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