Results 1 to 1 of 1

Thread: Analysis - c-translate of an integrable function is integrable

  1. #1
    Newbie
    Joined
    Jul 2015
    From
    N/A
    Posts
    14

    Analysis: c-translate of an integrable function is integrable

    I have difficulty proving this simple result: If $\displaystyle f\in \mathcal{R}[a,b]$, then $\displaystyle g(y):=f(y-c)\in \mathcal{R}[a+c,b+c]$ for $\displaystyle c\in \Bbb R$.
    I want to use Cauchy condition to prove this. (If you know other simple proofs, please tell me.)

    My attempt: Let $\displaystyle \varepsilon>0$. Since $\displaystyle f\in \mathcal{R}[a,b]$, there exists a partition of $\displaystyle [a,b]$, $\displaystyle P=\{a=x_0,x_1,\cdots,x_n=b\}$ such that $\displaystyle |U(f,P)-L(f,P)|<\varepsilon$. Pick $\displaystyle P_c=\{a+c=x_0+c,x_1+c,\cdots,x_n+c=b+c\}$. Then $\displaystyle |U(g,P_c)-L(g,P_c)|=|U(f,P_c)-L(f,P_c)|$. I have no idea what to do now. How can I compare this with $\displaystyle |U(f,P)-L(f,P)|$? Those are completely different partition points.

    Can someone also explain the arguments in this alternative proof?
    Last edited by ach4124; Sep 16th 2016 at 11:14 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. f & g Riemann integrable, show fg is integrable
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Feb 12th 2011, 09:19 PM
  2. Real Analysis: Riemann Integrable
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Dec 6th 2010, 05:51 AM
  3. Is a bounded integrable function square integrable?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 28th 2010, 06:26 PM
  4. Real Analysis: Prove BV function bounded and integrable
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Jan 17th 2010, 02:31 PM
  5. [SOLVED] f integrable implies f^2 integrable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: Jun 8th 2009, 10:53 PM

/mathhelpforum @mathhelpforum