# Riemann integrable function on [0, 1]

Consider a function fâ?¶[0,1]â?'[0,1] given by

f(x)={(1/q, if x=p/q,where p,qâ??N are coprime,

0, if x is irrational,

1, if x=0.

(a) Show that if xâ??[0,1] is rational, then f is not continuous at x.

(b) Show that L(f,P)=0 for any partition P of [0,1].

(c) Show that for any qâ??N the number of elements in the set

X(q)={xâ??[0,1]:f(x)â?¥1/q }

can be bounded by q(q+3)â?2. (Better bounds are possible, but not required.)

(d) Using that X(q) is finite, show that f is continuous at any irrational xâ??[0,1].

(e)Show that for any ?>0, we can find a partition P_? of [0,1] such that U(f,P_?)<?.

(f)Deduce that f is Riemann integrable and

â?«^1_0 f(x)dx=0.

https://brainmass.com/math/graphs-and-functions/riemann-integrable-function-on-0-1-395210

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