ANSWER
Since the set of value of the function is {−1,1} . then the function is bounded
∣f(x)∣=1 for all x∈[−1,1] .
In each interval (n+11,n1) (n∈N) the function is constant (=−1 or 1) , therefore it is continuous. At the points xn=n1 the function is discontinuous , because
limx→(2n−11)−f(x)=limx→(2n−11)−(−1)(2n−2)=1limx→(2n−11)+f(x)=limx→(2n−11)+(−1)(2n−3)=−1
and vice versa for points x2n . Note, that limn→∞xn=0, and limn→∞f(xn)=limn→∞(−1)n does not exist. Therefore x=0 is also a point of discontinuity of function.
Denote B the set of discontinuity of function. B={0,1,21,...,n1,...} and use
Definition .A set A⊂R has measure (Lebesgue) zero (m(A)=0) , if for all ϵ>0, there is a countable collection of open intervals {I0.I1,...,In,...} such that
A⊂⋃n=0∞In and . ∑n=0∞m(In)<ϵ .
If I=(α,β) , then m(I)=β−α.
Let A=B,I0=(−4ϵ,4ϵ),In=(n1−2n+2ϵ,n1+2n+2ϵ) . 0∈I0,n1∈In. So B⊂⋃n=0∞In ,
m(In)=2n+1ϵ(n=0,1,2,...) and ∑n=0∞m(In)=∑n=0∞2n+1ϵ=1−212ϵ=ϵ . Hence , m(B)=0.
Therefore, by the Riemann -Lebesgue Theorem, f is Riemann integrable on [0,1].
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