Re: Correct me if I'm wrong..

R. Colacitti wrote:

Correct me if I'm wrong...

1) If a function, f, is continuous on [a,b] then it ALWAYS has an
antiderivative for [a,b].

2) If a function, g, is integrable on [a,b] then the
definite integral of g from a to x ( where x is in (a,b] ) is ALWAYS
continuous.
both correct


Whare are some examples of functions that are integrable on a closed
interval, yet not continuous on that interval?
example 1...

f(x) = 0 for 0 <= x < 1/2 and f(x) = 1 for 1/2 <= x <= 1

example 2...

f(x) = 0 if x is irrational, f(x) = 1/q if x=p/q rational in lowest
terms


The other poster pointed out that your latter example is
Reimann-interable. How about the former?

Yes, it's Riemann-integrable. Otherwise, it would not have been an
example.

More generally, if _f_ is any bounded function from [a,b] into R with
only a finite or countable set of points at which _f_ is not continuous,
then _f_ is Riemann-integrable.

Best regards,

Jose Carlos Santos
.