Results 1 to 2 of 2

Math Help - well defined

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    154

    well defined

    Let f,g:R \rightarrow R be Lebesgue integrable functions. Let \phi(x,y)=f(x-y)g(y). Show that \phi is well defined.

    i am not sure what it means by a function is well defined. would someone tell me what i have to show to say that it is well defined? any help would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member Dark Sun's Avatar
    Joined
    Apr 2009
    From
    San Francisco, California
    Posts
    36
    Thanks
    1
    Well-defined means that for each input of a function, there is one output. For example, a circle is not a well-defined function.

    First, note that since f,g are Lebesgue integrable, they are well defined.

    Also, x=x',y=y' if and only if (x,y)=(x',y').

    Then, we have that since f,g are well-defined, then f(x-y)=f(x'-y'), on account of x-y=x'-y'. Similarly, g(y)=g(y').

    Hence  \phi (x,y)=f(x-y)g(y)=f(x'-y')g(y')=\phi (x',y'), and \phi is well-defined.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 17th 2011, 02:46 AM
  2. Which of the following is not defined?
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: October 12th 2009, 03:23 PM
  3. Well defined
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: September 23rd 2009, 12:03 AM
  4. Replies: 2
    Last Post: August 5th 2009, 11:20 AM
  5. well defined
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 22nd 2008, 02:06 PM

Search Tags


/mathhelpforum @mathhelpforum