Background I am trying to prove the following theorem. Let $f:[a,b]\to\mathbb{R}$ be a bounded function. If $c\in(a,b)$ then show that $f:[a,b]\to\mathbb{R}$ is Riemann Integrable on $[a,b]$ if $f$ is Riemann Integrable on both $[a,c]$ and $[c,b]$. Notation We use the following notations to simplify our discussion. The collection of all partitions on $[a,b]$ for the […]

If the left Riemann sum of a function over uniform partition converges, is the function integrable? To put the question more precisely, let me borrow a few definitions first. Pardon my use of potentially non-canon definitions of convergence. Given a bounded function $f:\left[a,b\right]\to\mathbb{R}$, A partition $P$ is a set $\{x_i\}_{i=0}^{n}\subset\left[a,b\right]$ satisfying $a=x_0\leq x_1\leq\cdots\leq x_n=b$. The […]

In many theorems about the Riemann-Stieltjes integral they required the hypothesis of $f$ to be bounded (for example: Suppose that $f$ is bounded in $[a,b]$, $f$ has only finitely many points of discontinuity in $I=[a,b]$, and that the monotonically increasing function $\alpha$ is continuous at each point of discontinuity of $f$, then $f$ is Riemann-Stieltjes […]

Yesterday I sat for my Real analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts. I tried it by choosing a partition $P_n=(0,\frac{1}{n},\frac{2}{n},\ldots,\frac{n-1}{n},1)$. But I was not able to show that $\displaystyle \lim_{n \to \infty} U(f,P_n)=\lim_{n \to \infty} L(f,P_n)=1$

Suppose $f:[0,1] \to (0,\infty)$ is a Riemann integrable function. Prove that the integral of the function from $0$ to $1$ is strictly positive. I have been trying to do this for awhile but I can’t seem to get it. Here is my thought process: If the function is riemann integrable, then its set of discontinuities […]

Prove that any function $f:[a,b]\rightarrow\mathbb{R}$ is Riemann integrable if it is bounded and continuous except a finite number of points. I will use the following criterion of Riemann integrability: A function $g$ is Riemann integrable on an interval $[a,b]$ if and only if for any $\epsilon>0$ there exist step functions $g_1$, $g_2$ such that $g_1(x)\le […]

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak. Give an example of a bounded function $f:[0,1] \to \mathbb{R}$ which is not Riemann Integrable, but is a derivative of some function $g$ on $[0,1]$.

Previously I posted this question asking for a review of the proof , but I realize the proof it’s wrong at all because I can’t have the continuity of $f$. So here is another attempt for the proof. If $f\in R$ on $[a,b]$ and $g$ is a monotonous function on $[a,b],$ then there exist $\epsilon […]

Consider one of the standard methods used for defining the Riemann integrals: Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\leq x_i$. Then if $$|\sigma|:=\max\{x_i-x_{i-1}|i=1,\cdots,n\},$$ which we shall call the norm of the subdivision, we define: $$\int_a^bf(x)dx:=\lim_{|\sigma|\to 0}\sum_{i=1}^nf(\xi_i)(x_i-x_{i-1}).$$ When one talks about the limit of a function $\lim_{x\to x_0}f(x)$, one has exactly one value […]

Let me ask right at the start: what is Riemann integration really used for? As far as I’m aware, we use Lebesgue integration in: probability theory theory of PDE’s Fourier transforms and really, anywhere I can think of where integration is used (perhaps in the form of Haar measure, as a generalization, although I’m sure […]

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