Principles of math analysis by Rudin, Chapter 6 Problem 7











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Suppose $f$ is a real function on $(0, 1]$ and $f in mathscr{R}$ on $[c,1]$ for every $c>0$. Define $int_0^1 f(x)dx=lim_{cto 0} int_c^1 f(x)dx$ if this limit exists (and is finite).



(a) If $f in mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.



(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.





This is Problem 7 of Chapter 6 in Principles of Mathematical Analysis by Rudin. For (a), I can prove the equation is correct but I am not sure what does 'definition agrees' mean? For (b), I have no idea.



Thank you in advance.










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  • 1




    For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
    – DanielWainfleet
    Nov 17 at 10:26












  • The above answer is more intuitive.
    – Tengerye
    Nov 20 at 9:02















up vote
1
down vote

favorite












Suppose $f$ is a real function on $(0, 1]$ and $f in mathscr{R}$ on $[c,1]$ for every $c>0$. Define $int_0^1 f(x)dx=lim_{cto 0} int_c^1 f(x)dx$ if this limit exists (and is finite).



(a) If $f in mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.



(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.





This is Problem 7 of Chapter 6 in Principles of Mathematical Analysis by Rudin. For (a), I can prove the equation is correct but I am not sure what does 'definition agrees' mean? For (b), I have no idea.



Thank you in advance.










share|cite|improve this question




















  • 1




    For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
    – DanielWainfleet
    Nov 17 at 10:26












  • The above answer is more intuitive.
    – Tengerye
    Nov 20 at 9:02













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Suppose $f$ is a real function on $(0, 1]$ and $f in mathscr{R}$ on $[c,1]$ for every $c>0$. Define $int_0^1 f(x)dx=lim_{cto 0} int_c^1 f(x)dx$ if this limit exists (and is finite).



(a) If $f in mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.



(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.





This is Problem 7 of Chapter 6 in Principles of Mathematical Analysis by Rudin. For (a), I can prove the equation is correct but I am not sure what does 'definition agrees' mean? For (b), I have no idea.



Thank you in advance.










share|cite|improve this question















Suppose $f$ is a real function on $(0, 1]$ and $f in mathscr{R}$ on $[c,1]$ for every $c>0$. Define $int_0^1 f(x)dx=lim_{cto 0} int_c^1 f(x)dx$ if this limit exists (and is finite).



(a) If $f in mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.



(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.





This is Problem 7 of Chapter 6 in Principles of Mathematical Analysis by Rudin. For (a), I can prove the equation is correct but I am not sure what does 'definition agrees' mean? For (b), I have no idea.



Thank you in advance.







calculus real-analysis integration






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edited Nov 17 at 3:52









qbert

21.7k32459




21.7k32459










asked Nov 17 at 3:12









Tengerye

477




477








  • 1




    For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
    – DanielWainfleet
    Nov 17 at 10:26












  • The above answer is more intuitive.
    – Tengerye
    Nov 20 at 9:02














  • 1




    For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
    – DanielWainfleet
    Nov 17 at 10:26












  • The above answer is more intuitive.
    – Tengerye
    Nov 20 at 9:02








1




1




For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
– DanielWainfleet
Nov 17 at 10:26






For (b) suppose, for $nin Bbb N,$ that $int_{1/(n+1)}^{1/n}f(x)dx=(-1)^n/n .$ Suppose that when $xin [1/(n+1),1/n]$ then $f(x)leq 0$ if $n$ is odd, while $f(x)geq 0$ if $n$ is even.
– DanielWainfleet
Nov 17 at 10:26














The above answer is more intuitive.
– Tengerye
Nov 20 at 9:02




The above answer is more intuitive.
– Tengerye
Nov 20 at 9:02










1 Answer
1






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oldest

votes

















up vote
2
down vote



accepted










$b$) Using the well known integral
$$
int_{1}^infty frac{sin x}{x}mathrm dx
$$

which converges conditionally, we reflect reflect everything to near $0$ by sending
$$
xto 1/x
$$

and find
$$
int_{1}^inftyfrac{sin x}{x}mathrm dx=
int_0^1frac{sin(1/y)}{y}mathrm dy
$$



For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
$$
int_0^1 f(x)mathrm dx
$$

is the same as the number
$$
lim_{cto 0^+}int_c^1f(x)mathrm dx
$$






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote



    accepted










    $b$) Using the well known integral
    $$
    int_{1}^infty frac{sin x}{x}mathrm dx
    $$

    which converges conditionally, we reflect reflect everything to near $0$ by sending
    $$
    xto 1/x
    $$

    and find
    $$
    int_{1}^inftyfrac{sin x}{x}mathrm dx=
    int_0^1frac{sin(1/y)}{y}mathrm dy
    $$



    For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
    $$
    int_0^1 f(x)mathrm dx
    $$

    is the same as the number
    $$
    lim_{cto 0^+}int_c^1f(x)mathrm dx
    $$






    share|cite|improve this answer

























      up vote
      2
      down vote



      accepted










      $b$) Using the well known integral
      $$
      int_{1}^infty frac{sin x}{x}mathrm dx
      $$

      which converges conditionally, we reflect reflect everything to near $0$ by sending
      $$
      xto 1/x
      $$

      and find
      $$
      int_{1}^inftyfrac{sin x}{x}mathrm dx=
      int_0^1frac{sin(1/y)}{y}mathrm dy
      $$



      For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
      $$
      int_0^1 f(x)mathrm dx
      $$

      is the same as the number
      $$
      lim_{cto 0^+}int_c^1f(x)mathrm dx
      $$






      share|cite|improve this answer























        up vote
        2
        down vote



        accepted







        up vote
        2
        down vote



        accepted






        $b$) Using the well known integral
        $$
        int_{1}^infty frac{sin x}{x}mathrm dx
        $$

        which converges conditionally, we reflect reflect everything to near $0$ by sending
        $$
        xto 1/x
        $$

        and find
        $$
        int_{1}^inftyfrac{sin x}{x}mathrm dx=
        int_0^1frac{sin(1/y)}{y}mathrm dy
        $$



        For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
        $$
        int_0^1 f(x)mathrm dx
        $$

        is the same as the number
        $$
        lim_{cto 0^+}int_c^1f(x)mathrm dx
        $$






        share|cite|improve this answer












        $b$) Using the well known integral
        $$
        int_{1}^infty frac{sin x}{x}mathrm dx
        $$

        which converges conditionally, we reflect reflect everything to near $0$ by sending
        $$
        xto 1/x
        $$

        and find
        $$
        int_{1}^inftyfrac{sin x}{x}mathrm dx=
        int_0^1frac{sin(1/y)}{y}mathrm dy
        $$



        For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
        $$
        int_0^1 f(x)mathrm dx
        $$

        is the same as the number
        $$
        lim_{cto 0^+}int_c^1f(x)mathrm dx
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 17 at 3:50









        qbert

        21.7k32459




        21.7k32459






























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