Finding if a function is Riemann Integrable

**tonio** · October 31st 2009, 11:10 PM

Originally Posted by xyz

Well, going with the hint you could argue that since f(x) is continuous in $[\theta,1]$ , for any $\theta>0$ , then f is uniformly continuous there, so $\forall\,\, \epsilon > 0\,\,\exists\,\,\delta>0\,\,s.t.\,\,|x-y|<\delta\,\Longrightarrow |f(x)-f(x)|<\epsilon$ , and then: taking a partition P with n points of $[\theta,1]$ s.t. $\max \{x_i-x_{i-1} /\,[x_{i-1},x_i ]\in P\}<\frac{\delta}{n\epsilon}$ , we get $U(f,P)-L(f,P)<\epsilon$ .
Of course, we still need to deal with $[0,\theta)$ , but this is not problem: the difference $\theta-0=\theta.$ can be made as little as wanted.

Now, not going with the hint is much simpler and shorter, but perhaps you guys haven't yet studied this: f is Riemann integrable there since it is bounded and the set of points of discontinuities of f in [0,1] (which is one single point) has (Borel-Lebesgue) measure zero.

Tonio

**xyz** · November 2nd 2009, 07:11 PM

As seen above is the graph of the function that has to be proven, it is riemann integrable. So this is how i answered. Please let me know if it is correct --

Thread: Finding if a function is Riemann Integrable

LinkBack

Thread Tools

Display

Finding if a function is Riemann Integrable

Similar Math Help Forum Discussions

Riemann integrable function?

Finding Riemann Integrable Function - Please help!

Is there a function f = g' but not Riemann Integrable

Riemann Integrable Function

Riemann Integrable Function

Search Tags