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Let f:[0,1]——>R be define by f(x)= x if x belong to rational number and 0 if...

Let f:[0,1]——>R be define by f(x)= x if x belong to rational number and 0 if x belong to irrational number and let g(x)=x

(a) prove that for all partitions P of [0,1],we have U(f,P)=U(g,P).what does mean about U(f) and U(g)?

(b)prove that U(g) greater than or equal 0.25

(c) prove that L(f)=0

(d) what does this tell us about the integrability of f ?

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Answer #1

a. Let be any partition of [0,1]. Clearly

Which gives that U(f)=U(g)

b. We have for any partition P, now we take the partition 0<0.5<1,

then,

Hence

c. Let

Since irrationals are dense, we can find an irrational number in , Hence

which is true for all partition P. Hence L(f)=0

d. we have  , hence f is not integrable

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