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

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What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?
 

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  • #2
HallsofIvy
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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.
 
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or [tex]f(x)=\left| x\right|[/tex]
 
  • #4
HallsofIvy
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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.
 
  • #5
mathwonk
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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.
 
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[tex]V_{3}[\tex]
 
  • #7
arildno
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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]
 
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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.
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.
 
  • #9
HallsofIvy
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Yes, you are right. That was an error on my part.
 

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