# Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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### Conditions for applying the second fundamental theorem of calculus with gauge integrals

I was thinking about this question while walking home today and can't seem to prove or come up with a counterexample myself. Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function, $f(x),$ ...
53 views

### Are bound functions always Henstock-kurzweil integrable?

Is there any function $f:[\alpha,\beta]\rightarrow\mathbb{R}$ that is bound but not Henstock-kurzweil integrable? I assume such a function would have to be horrendously discontinuous but I am unable ...
56 views

### What are the necessary and sufficient conditions for a function to be Henstock–Kurzweil integrable?

I recently stumbled upon Lebesgue’s criterion for Riemann integrability. It didn't take very long until I found this result quite intuitive. I then began studying the Henstock–Kurzweil integral. Very ...
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### Henstock-Kurzweil integral of sin(x)/x by directly using explicit gauge.

I am trying to show that the Henstock-Kurzweil integral of sin(x)/x from zero to infinity is $\pi/2$. In order to do this, I wish to construct an explicit gauge and use Cousin's lemma to find a Perron ...
32 views

### Definition of Henstock integral function over a set

I understand the definition of Henstock integrable function on $[a, b]$, i.e., $f$ is Henstock-Kurzweil integrable on $[a, b]$ if there is $A \in \mathbb{R}$ with property for every $\varepsilon>0$...
107 views

### Is the derivative of $x \cos\left(\frac{\pi}{x}\right)$ integrable?

Let $F:[0,1]\rightarrow \mathbb{R}$ defined by $$F(x)= \left\{ \begin{array} .x\cdot \cos\left(\frac{\pi}{x}\right), &\textrm{if } x\in[0,1] \\ 0, &\textrm{ if x=0} \end{array} \right.$$ ...
74 views

### generalized Riemann Integrability of $f\cdot g$

Let $\mathcal{R}([a.b])$ the set of all Riemann-integrable functions in $[a,b]$. Let $\mathcal{R}^{*}([a,b])$ the set of all Generalized Riemann-Integrable functions in $[a,b]$ (I'm talking about the ...
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569 views

### Mistake in Bartle's proof of Hake's Theorem?

Here is Bartle's proof of Hake's Theorem found in "A Modern Theory of Integration". I think there is a mistake in the highlighted line: The Theorem: $f:[a,b]\to \mathbb{R}$ is gauge integrable if and ...
164 views

### Proof that if $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$

I am asking for a self contained proof of this assertion: If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$. The integral in question is the ...
876 views

### Integrability: Neither improper Riemann nor Lebesgue but Henstock-Kurzweil

Can you think of a function that is neither improper Riemann nor Lebesgue integrable, but is Henstock-Kurzweil integrable? I'd like to put a bounty on this question, but my reputation is not nearly ...