1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integrability question

  1. Dec 23, 2011 #1
    The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval.

    Or basically if it is increasing or decreasing on the interval it is integrable on that interval

    This makes sense, however this theorem seems to obvious because obviously if a function is countinuous on a closed interval it will be integrable on that interval whether or not its increasing or not...
    So my question is... what is a non monotonic function..? would that be a function with discountinuities?
     
  2. jcsd
  3. Dec 23, 2011 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It doesn't have to be discontinuous. For example f(x) = xsin(1/x) is continuous if you define f(0) = 0, but it isn't monotonic on any closed interval containing 0. A discontinuous example is the "salt and pepper" function g(x) = 1 if x rational and 0 if x irrational, which is not monotonic on any interval.
     
  4. Dec 23, 2011 #3
    Sorry, I dont see how f(x) = xsin(1/x) is defined at x = 0...?
     
  5. Dec 23, 2011 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The definition of the function I am suggesting is$$
    f(x) = \begin{cases} \frac 1 x\sin(x)&x \neq 0\\
    0 & x = 0\end{cases}$$ It is defined to be 0 when x = 0.
     
  6. Dec 23, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    f(x)=x^2 on [-1,1] is nonmonotonic. It is continuous. It's also integrable. What's the question again? I think Miike012 might be confusing f monotonic -> f integrable (which is true) with f not monotonic -> f not integrable (which is false).
     
    Last edited: Dec 23, 2011
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integrability question
  1. Integral Questions (Replies: 2)

  2. Integration question (Replies: 2)

  3. Integral Question (Replies: 5)

  4. Integral Question (Replies: 8)

  5. Integration Question (Replies: 4)

Loading...