0
$\begingroup$

Can someone please explain why the function $$f(x) = \frac{1}{x}$$ is not integrable in the open interval $(0,1)$ but indeed integrable in the close form $[0,1]$? Thanks.

$\endgroup$
  • 7
    $\begingroup$ Why do you think $\tfrac{1}{x}$ is integrable on $[0,1]$? $\endgroup$ – StackTD Jun 16 '16 at 17:06
  • 1
    $\begingroup$ The Riemann integral isn't even defined on open intervals.... $\endgroup$ – MathematicsStudent1122 Jun 16 '16 at 18:00
0
$\begingroup$

Note that $$\lim_{a\to 0}\ln a = -\infty$$

From this you can conclude that $f$ is neither integrable on the open nor on the closed interval.

$\endgroup$
0
$\begingroup$

The key concept is that a single point has length (measure) zero, so adding or subtracting the extrema to an interval does not change the integral. In your case all the two integrals does not exists.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.