Checking for integrability on a half-open interval

schniefen

Problem Statement
For a bounded, continuous and monotonous function on a half-open interval $(a,b]$, how does one check if the function is integrable? (specifically Darboux integrable)
Relevant Equations
My definition of Darboux integrable: $U(f,P)-L(f,P)<\epsilon$ for all $\epsilon>0$
For a closed interval $[a,b]$ I have learned that $U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}$ where $N$ is the number of subintervals of $[a,b]$ (if $f$ is monotonically decreasing, change the numerator of the fraction to $f(a)-f(b)$). However, if the interval is half-open, then $f(a)$ is no longer defined. How does one go about this issue?

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PAllen

I assume you mean monotonic, not monotonous, though a dubious case can be made for synonimity.

PAllen

More seriously, I assume you have been presented with a definition of the integral for cases other than monotonic. Try working from that definition.

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schniefen

Yes, I mean monotone of course. The only definition of Darboux integrability I’ve been given is the epsilon one above, although this can be written somewhat differently. And when I check the integrability for a certain bounded, continues and monotone function on a closed interval I make use of the fact that the difference between the upper and lower Riemann sums create a column of height $f(b)-f(a)$ and width $(b-a)/N$. Then finding an $N$ for all $\epsilon>0$ is quite easy.

PAllen

Just think about how to extend the argument slightly. You have monotonicity and continuity given.

schniefen

Could one write $\lim\limits_{x \to a^+}f(x)$ instead of $f(a)$? Is this limit evaluable?

pasmith

Homework Helper
Problem Statement: For a bounded, continuous and monotonous function on a half-open interval $(a,b]$, how does one check if the function is integrable? (specifically Darboux integrable)
Relevant Equations: My definition of Darboux integrable: $U(f,P)-L(f,P)<\epsilon$ for all $\epsilon>0$

For a closed interval $[a,b]$ I have learned that $U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}$ where $N$ is the number of subintervals of $[a,b]$ (if $f$ is monotonically decreasing, change the numerator of the fraction to $f(a)-f(b)$). However, if the interval is half-open, then $f(a)$ is no longer defined. How does one go about this issue?
Don't use that formula: it assumes a closed interval and is therefore not appropriate to a half-open interval.

In fact, $f$ will be Darboux integrable on $(a, b]$ because it is bounded and continuous; it is not necessary that it be monotonic (for example, $\sin (x^{-1})$ is integrable on $(0,1]$).

The starting point is the definition of Darboux integrability:

A function $f : (a,b] \to \mathbb{R}$ is integrable if and only if for every $\epsilon > 0$ there exists a partition $P$ such that $U(f,P) - L(f,P) < \epsilon$.

The key observation is that as $f$ is bounded the contribution to $U(f,P) - L(f,P)$ from the first subinterval is bounded above by $K\delta$ where $\delta$ is the width of the subinterval and $K \geq 0$ does not depend on the choice of partition. This can be made arbitrarily small by suitable choice of $\delta$. As $f$ is continuous and thus integrable on the closed bounded interval $[a + \delta, b]$, the contribution to $U(f,P) - L(f,P)$ from $[a + \delta, b]$ can be made arbitrarily small by suitable choice of subintervals.

schniefen

The key observation is that as $f$ is bounded the contribution to $U(f,P) - L(f,P)$ from the first subinterval is bounded above by $K\delta$ where $\delta$ is the width of the subinterval and $K \geq 0$ does not depend on the choice of partition. This can be made arbitrarily small by suitable choice of $\delta$. As $f$ is continuous and thus integrable on the closed bounded interval $[a + \delta, b]$, the contribution to $U(f,P) - L(f,P)$ from $[a + \delta, b]$ can be made arbitrarily small by suitable choice of subintervals.
What is $K$? If one were to prove that a piece-wise continuous function is integrable, say $f(x)=\tan{(x)}$ on $[0,\pi/3)$ and $f(x)=\sin{(x)}$ on $(\pi/3,\pi]$, does one simply show that we can treat both intervals as closed intervals?

PAllen

What does bounded mean? That should give you a clue to what K might be.

"Checking for integrability on a half-open interval"

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