Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Examples where it's Riemann integrable but no derivative exists at pts

  1. Nov 17, 2008 #1
    What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?
     
  2. jcsd
  3. Nov 17, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The function f(x)= 0 if [itex]x\le 0[/itex], 1 if x> 0 is integrable but has no derivative at x= 0. More generally, if f(x) has finite "jump" discontinuities at some points, it is still Riemann integrable but is not differentiable at those points.
     
  4. Nov 18, 2008 #3
    or [tex]f(x)=\left| x\right|[/tex]
     
  5. Nov 18, 2008 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Those two examples also have the property that while [itex]F(x)= \int f(t)dt[/itex] is defined, F(x) itself has no dervative at x= 0.
     
  6. Nov 19, 2008 #5

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    there are functions which are continuous everywhere hence integrable, but differentiable nowhere. perhaps the famous dirichlet function which equals zero at irrationals and 1/q at p/q is even differentiable nowhere. since it is continuous a.e. it is integrable.
     
  7. Nov 19, 2008 #6
    [tex]V_{3}[\tex]
     
  8. Nov 20, 2008 #7

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    An example of a continuous (and hence integrable) function that is nowhere differentiable is the Weierstrass function:
    [tex]F(x)=\sum_{n=0}^{\infty}\frac{\sin((n!)^{2}x)}{n!}[/tex]
     
  9. Nov 20, 2008 #8
    I'm not sure about this. Isn't [itex]F(x)= \int_0^x |t|dt[/itex] differentiable at 0? It is the piecewise function given by F(x)=x^2 for x>0 and F(x)=-x^2 for x>0 and F(0)=0.
     
  10. Nov 20, 2008 #9

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you are right. That was an error on my part.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Examples where it's Riemann integrable but no derivative exists at pts
  1. Riemann Integration (Replies: 4)

Loading...