Deduce the function is riemann integrable

Question

Solve the following:

Q: Consider a function f:[0,1]→[0,1] given by

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

          0, if x is irrational

          1, if x=0.}

a Show that if x∈[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 q∈N the number of elements in the set

X(q)={x∈[0,1]:f(x)≥1/q }
can be bounded by q(q+3)⁄2. (Better bounds are possible, but not required.)

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

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

f Deduce that f is Riemann integrable and

₀¹∫f(x)dx=0