# integrability of function in open interval

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.

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

## 2 Answers

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.

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.