Re: integrability of f(x)

Ardi wrote:

Can someone help me get started solving this problem:

let f(x) be defined as follows:

f(x) = 1, x=1, 1/2, 1/3, 1/4, 1/5, ..........
f(x) = 0 otherwise

show that f(x) is integrable on [0,1] and what is the
value of the integral?

Riemann integrable? Lebesgue integrable? Henstock-Kurzweil
integrable? Also, what is the specific definition that you
are using for your integral (e.g. in the case of the Riemann
integral, are you using a limit of Riemann sums, the equality
of upper and lower Darboux integrals, the finite Jordan content
of those points where the oscillation is greater than or
equal to an arbitrarily specified positive number, etc.)
and what proof constraints are you working under? Since the
function obviously has at most countably many discontinuities,
it's immediate that its discontinuities form a set of measure
zero, and hence the function is Riemann integrable. However,
the fact that you can't get started suggests that the easiest
and most obvious way of attacking this problem isn't available
to you. Thus, you should probably be a little more specific
in saying what you want.

Sorry about the mild flame, but if you can give some more
context for your question and give some idea of your background,
I think you'll find others more willing to help. As for me,
I'm going to bed now, but maybe someone else can give you
some hints after you clear up the issues I raised.

Dave L. Renfro

.

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