Question #346388

Show that the function f defined on [1,0] by f(x) = (-1)n-1 for 1/n+1 <x< = 1/n (for n= 1,2,3,...) is integrable on [0,1]


1
Expert's answer
2022-06-01T13:34:43-0400

ANSWER

Since the set of value of the function is {1,1}\left \{ -1,1 \right \} . then the function is bounded

f(x)=1|f(x)| = 1 for all x[1,1]x\in [-1,1] .

In each interval (1n+1,1n)(\frac {1}{n+1}, \frac {1}{n}) (nN)(n\in \N) the function is constant (=1(=-1 or 1)1) , therefore it is continuous. At the points xn=1nx_{n}=\frac{1}{n} the function is discontinuous , because

limx(12n1)f(x)=limx(12n1)(1)(2n2)=1limx(12n1)+f(x)=limx(12n1)+(1)(2n3)=1\lim_{x\rightarrow (\frac {1}{2n-1})^{-} }f(x) = \lim_{x\rightarrow (\frac{1}{2n-1})^{-} } (-1)^{ (2n-2) }=1 \\\lim_{x\rightarrow (\frac {1}{2n-1})^{+} }f(x) = \lim_{x\rightarrow (\frac{1}{2n-1})^{+} } (-1)^{ (2n-3) }=-1

and vice versa for points x2nx_{2n} . Note, that limnxn=0,\lim_{n\rightarrow \infty }x_{n}=0, and limnf(xn)=limn(1)n\lim_{n\rightarrow \infty}f(x_{n})=\lim_{n\rightarrow \infty}(-1)^{n} does not exist. Therefore x=0x=0 is also a point of discontinuity of function.

Denote BB the set of discontinuity of function. B={0,1,12,...,1n,...}B= \left \{ 0,1,\frac{1}{2} ,..., \frac{1}{n},...\right \} and use

Definition .A set ARA\subset \R has measure (Lebesgue) zero (m(A)=0)(m(A)=0) , if for all ϵ>0,\epsilon>0, there is a countable collection of open intervals {I0.I1,...,In,...}\left \{ I_{0} .I_{1},..., I_{n},...\right \} such that

An=0InA\subset\bigcup_{n=0}^{\infty} I_{n} and . n=0m(In)<ϵ\sum_{n=0}^{\infty} m(I_{n} )<\epsilon .

If I=(α,β)I=(\alpha,\beta) , then m(I)=βα.m(I)=\beta-\alpha.

Let A=B,I0=(ϵ4,ϵ4),In=(1nϵ2n+2,1n+ϵ2n+2)A=B, I_{0}=(-\frac{\epsilon}{4}, \frac{\epsilon}{4}), I_{n}=(\frac{1}{n}-\frac{\epsilon}{2^{n+2}}, \frac{1}{n}+\frac{\epsilon}{2^{n+2}}) . 0I0,1nIn.0\in I_{0}, \frac{1}{n}\in I_{n}. So Bn=0InB\subset\bigcup_{n=0}^{\infty} I_{n} ,

m(In)=ϵ2n+1(n=0,1,2,...)m(I_{n})=\frac{\epsilon}{2^{n+1}} (n=0,1,2,... ) and n=0m(In)=n=0ϵ2n+1=ϵ2112=ϵ\sum_{n=0}^{\infty} m(I_{n} )=\sum_{n=0}^{\infty} \frac{\epsilon}{2^{n+1}} =\frac{\frac{\epsilon}{2}}{1-\frac{1}{2}} =\epsilon . Hence , m(B)=0.m(B)=0.

Therefore, by the Riemann -Lebesgue Theorem, ff is Riemann integrable on [0,1][0,1].



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