What Is an Example of a Nonintegrable Bounded Function?

[matrix_204]

could someone give me an example of a function that is bounded but is nonintegrable?


i need to know what a nonintegrable function bounded on [a,b] is as said in my preperation file for a test? urgent help needed

 

[dextercioby]

How about
f:(0,1)\rightarrow R,

f(x)=0,x\in ((0,1) \cap R-Q)

f(x)=1,x\in ((0,1) \cap Q)

Daniel.

 

[matrix_204]

yup thnx, i forgot we did an example of dirichilet function in class, this function is an example of a lot of things used in calculus, lol,

 

[dextercioby]

Always remember that continuity is a necessary condition for integrability...

Daniel.

 

[AKG]

Continuity is a sufficient condition for integrability, not a necessary one. A function that is discontinuous on a set of points of measure 0 is integrable, and vice versa (i.e. this gives a necessary and sufficient condition). Clearly, a continuous function is discontinuous on an empty set which of course has measure 0, so it is integrable. The example you gave is discontinuous on (0, 1), a set that doesn't have measure 0, which is why f is not integrable. Of course, this also depends on how you define integration and integrability.

 

[dextercioby]

Are u talking about Lebesgue,or Riemann integrability...?

Daniel.

 

[HallsofIvy]

dextercioby: The function f(x)= 0 if x< 0; 1 if 0< x< 1; 0 if x> 1 is (Riemann) integrable over any interval but is not continuous at 0 and 1.

The function: f(x)= 0 if x is rational; 1 if x is irrational is (Lebesque) integrable over any interval but is not continuous anywhere.

 

[Hurkyl]

AKG's right, I'm pretty sure. IIRC, A bounded function is Riemann integrable over a compact set iff it's discontinuous on a set of measure zero.

 

[dextercioby]

Got it.Thank you.

Daniel.