4
votes
4answers
116 views

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
3
votes
0answers
30 views

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 $(a, b)$, ...
1
vote
1answer
44 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
1
vote
1answer
20 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
4
votes
1answer
74 views

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 ...
2
votes
1answer
25 views

$ \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 ...
0
votes
2answers
48 views

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 ...
1
vote
0answers
43 views

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 ...
1
vote
0answers
27 views

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 ...
1
vote
2answers
37 views

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 ...
0
votes
2answers
31 views

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 ...
1
vote
1answer
38 views

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 ...
1
vote
2answers
77 views

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 $f$ ...
1
vote
1answer
33 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
1
vote
1answer
33 views

Show that an integral can be made as small as possible.

Consider a function $\mu(s)$ satisfying the following properties: $\mu(s) \in C^0((0,+\infty))$, $\mu(s) > 0$ and $\mu(s)$ is increasing in $s \in (0,+\infty)$, $\displaystyle \int_0^1 ...
1
vote
2answers
48 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
1
vote
1answer
91 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
1
vote
1answer
124 views

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 ...
1
vote
1answer
46 views

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 ...
1
vote
2answers
47 views

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 ...
0
votes
0answers
38 views

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 ...
9
votes
1answer
330 views

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 ...
1
vote
0answers
52 views

Proof that order of integration does not matter for non-continuous functions

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 ...
0
votes
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, ...
6
votes
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 ...
0
votes
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 ...
1
vote
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 $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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 ...
11
votes
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 ...
1
vote
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 ...
3
votes
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 $I(0) = \frac{\pi}{2}$. ...
4
votes
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 ...
3
votes
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 ...
2
votes
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 ...
0
votes
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. ...
2
votes
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 ...
3
votes
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 ...
2
votes
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 $X \subseteq \mathbb{R}^n$, ...
2
votes
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] ...
1
vote
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
3
votes
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 ...
4
votes
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 ...
2
votes
1answer
185 views

Continuity of metric space of integrals of continuous functions

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 ...