Tagged Questions

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If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
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Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
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Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
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$\int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2)\; dt = \frac{(-1)^n}{c}, \text{ where } \sqrt{n\pi} \leq c \leq \sqrt{(n+1)\pi}.$

The following is a problem from Apostol Vol 1 Calculus from the section: Continuity. Since Differentiation hasn't been introduced yet, the objective is to solve it without direct reference to ...
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integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0$. Show that $f$ must be constant. Well, it's clear ...
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Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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Check my answer - show a function is integrable and find the integral

Let $Q =[0,1]$x$[0,1]$. Let $f: Q \to \mathbb R$ defined as such: if $(x,y) \in \mathbb Q$x$\mathbb Q$ then $f(x,y)=\frac{1}{n_1}+\frac{1}{n_2}$ where $x=\frac{m_1}{n_1}$ and $y=\frac{m_2}{n_2}$ are ...
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How to prove that this function is continuous?

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on the rectangle $R=[a,b] \times [c,d]$, prove that the function $g(x) := \int\limits_{c}^{d} f(x,y) dy$ is continuous on $[a,b]$. Thanks in ...
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If the lower Darboux integral is zero, does the function equal zero for every x?

So say I let a function $f$ be a continuous function on the interval $[a,b]$ such that $f(x)$ is greater than or equal to $0$ for every $x \in [a,b]$. If $L(f) = 0$, does $f(x)=0$ for every $x$ in ...
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Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$\int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
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Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
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Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big]$$
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Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
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Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
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What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
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Confusion about validity of integration by substitution method

Suppose $$f(x,y) = \int g(x,y,t)\, dt,$$ and I wish to do the integral over $t$ by setting $$t = h(x,y,\tau).$$ As an example, suppose $$h(x,y,\tau) = \frac{\tau}{\cos x} -\tan y.$$ If we ...
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Proving the existence of a point with a certain property for a continuous function

Let $f:[0,1]\to\mathbb{R}$ a continuous function and $\int_0^1xf(x)dx=0$. Show that there exists a point $c\in(0,1)$ so that $f(c)=(\int_c^1f(x)dx)^2$. As a potential solution, I tried assuming that ...
For continuous functions of several variable, by Young's theorem the order of integration (i.e. "antiderivation") does not matter so that for instance $$\int \left(\int f(x,y)~ dx\right) ~dy = \int ... 0answers 54 views f \in \mathcal{R}(\alpha) on [a,b], then \exists P_n s.t. \lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0. Assume f \in \mathcal{R}(\alpha) on [a,b], and prove that there are polynomial P_n such that \lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0. This is what I have, ... 3answers 881 views integral from zero to zero it seems obvious that this integral is zero and so is the limit but what theorem we are using here? I see it's connected to Riemann sums with an interval=zero Right ? The function \mathrm{f} is ... 1answer 65 views integral = zero Let f be a continuous function in [a,b] For every continuous function g(x) : \displaystyle\int_{a}^{b}f(x)\cdot g(x)\;\mathrm dx=0 We need to prove that f(x)=0. I thought about proving by ... 1answer 124 views f is bounded and continious \Rightarrow the convolution integral \int f(\tau)g(x-\tau)\text{ d}\tau is bounded and continuous Let g\in L^1(\mathbb{R}^n) and f:\mathbb{R}^n\to\mathbb{R} be bounded and continuous. Why is the convolution integral$$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$... 1answer 125 views Proving that f(x)\geq 0 on [0,1) when f(x) is continuous and when the Darboux/Riemann integral of f(x) is greater than 0. Suppose that f(x) is continuous and defined on [0,1). Also, suppose that the Riemann/Darboux integral \int_a^b f(x)\geq 0 on [0,1) for any partition . Show that f(x)\geq 0 for all x\in ... 1answer 61 views Continuous function in two variables Let's have a look to the function F(a,b)=\int_0^{2\pi}{\sqrt{a^2\sin^2(t)+b^2\cos^2(t)}dt}, then this function describes the length of the arc of an ellipse (a,b>0). Can we see when this ... 3answers 87 views Continuity and Integration Let f,g:[0,1]\rightarrow{\mathbb{R}} be continuous functions such that f(0)=0, \int_0^1 f(x) \,dx=0, g(0)=0, \int_0^1 g(x) \,dx=1. Then I am having trouble showing that ... 2answers 107 views Introduction to Analysis: The Riemann Integral The following is a problem from Arthur Mattuck's book, "Introduction to Analysis." Page 265. Assume f(x) integrable on I. Prove F(x) = \int_a^x f(t)\,dt is continuous on I How would I ... 1answer 182 views example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle Construct a nontrivial example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle. A (trivial) example would be to rede fine a nice ... 3answers 374 views Show that if f is continuous on [a,b], then there is a Riemann sum that equals the integral Specifically, I mean a Riemann sum of f over [a,b] that is equal to \int_a^b f. I encountered this question on my analysis exam, but I was unable to answer it. I thought it was an interesting ... 1answer 307 views if f is continuous and absolutely integrable \lim\limits_{x\to\infty}f(x)=0 Prove that if f(x) is continuous and absolutely integrable on [a,\infty) then \lim\limits_{x\to\infty}f(x)=0. I tried proving it in the following way: First we need to prove the existence of ... 1answer 268 views Showing that \Omega is of class C^1 I have done a lot in this problem, but unfortunately it is not enough to solve it, answers or hints are very welcome. Let B be a rectangle in \mathbb R^2 and consider \varphi\colon ... 1answer 32 views Continuity of an integral with respect to one variable Let V\subseteq \mathbb{R}^n and f:V\to\mathbb{R}^n. Consider the function$$g(x_1,x_2,...,x_n) = \int_{x_2}^{x_1} {f(t,x_2,...,x_n)dt}$$on V. What conditions will I need to conclude that g is ... 2answers 399 views Continuity of integral function How to show that the following function is right continuous at 0 (that is, when a\to0+): I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx? I know that Lebesgue integral I(0) = \frac{\pi}{2}. ... 1answer 201 views Limit with integral or is this function continuous? Hello I need to show one identity and one limit. I am having problems with it. notation: x_i is i-th coordinate of x B(x,r) ball with center x and radius r S(x,r) sphere with center ... 2answers 117 views Berkeley exam summer '79, sequence of continuous functions, integral, convergence I've recently been browsing some Berkeley exams and I'm particularly interested in Problem 19 here. Let {f_n} be a sequence of continuous real functions deﬁned on [0,1] such that \int_0^1 ... 1answer 92 views Sequence of continuous functions, integral, series convergence Let f_k be a sequence of continuous functions on [0,1] such that \int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx for all n \in \mathbb{N}. Is \sum _{k=1} ^{\infty}f_k(x) convergent? Could ... 1answer 66 views Continuous function involved with integrals and limit Let f:[0,\infty)\rightarrow R be a continuous function such that for all A>0 the integral \int_{A}^{\infty}\frac{f(t)}{t}dt converges. Suppose that 0<a<b. Show that a. ... 1answer 133 views Riemann Integration, Uniform Continuity Let f\colon\mathbb{R}\to\mathbb{R} be a bounded function that is integrable on any bounded subinterval of \mathbb{R} (but f is not necessarily continuous). Let T be a positive constant. We ... 2answers 49 views Continous value of integrals Let$$f:(a,b] \rightarrow \mathbb{R} $$if the riemann Integral$$ F(c):= \int_c^b f(x) dx $$for all$$ c \in (a,b) $$exists and the improper integral$$ F(a)=\int_a^b f(x) dx$$exists too. Does this ... 1answer 112 views Continuity of \max of Lebesgue integral Let m be a probability measure on Z \subseteq \mathbb{R}^p, so that m(Z)=1. Consider a locally bounded f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}, with X \subseteq \mathbb{R}^n, ... 1answer 57 views Show f(x) = 0 given continuity and integrals. I'm having some trouble with the following problem. I've listed what I've used so far. (Edit: Made a mistake with the second integral.) The problem is as follows: If a function f: [0,1] ... 3answers 224 views Continuity of the lebesgue integral How does one show that the function, g(t) = \int \chi_{A+t} f  is continuous, given that A is measurable, f is integrable and A+t = \{x+t: x \in A\}. Any help would be appreciated, thanks 1answer 1k views f bounded on [a,b] with one or finite discontinuities implies f Riemann-integrable. I have two problems: Prove that if f is bounded on [a, b] and has exactly one discontinuity in [a, b] then f is Riemann-integrable on [a, b]. Prove that if f is bounded on [a, b] and ... 1answer 154 views f continous at x_0 ⇒\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0) The function f:ℝ→ℝ is continuous on x_0\inℝ. Prove using the definition of a Darboux Integral that$$\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0) I'm a first grade math student following an ...
Let $R$ be the real line with the standard metric $d:R \times R \to R$ be defined by $d(x,y) = |x-y|$. Let $X$ be the set of continuous functions $f:[a,b] \to R$ of an arbitrary closed interval ...