1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that no such functions exist

  1. Nov 6, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that there do not exist functions ##f## and ##g## with the following property:
    $$(\forall x)(\forall y)(f(x+y) = g(x) - y)$$

    2. Relevant equations
    NA

    3. The attempt at a solution
    Here is some information I have found out about ##f## and ##g## if we suppose they exist:
    ##f(x +0) = f(x) = g(x) = g(x)-0## for all x, so ##f## and ##g## are equal. Hence, ##f(x+y) = f(x) - y## for all ##x## and ##y##. Thus, ##f(y) = f(0) - y##, so ##f## is a linear function. Any suggestions as to what kind of values I have to substitute to arrive at a contradiction from here on? Thanks!
     
  2. jcsd
  3. Nov 6, 2015 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    What you have done already is very good! The problem is that what you are trying to proven simply isn't true! Yes, setting y= 0 we have f(x)= g(x). And taking x= 0, f(y)= f(0)- y. Since f(0) is a constant, write f(x)= A- x where the number A is to be determined. You can't prove that such f and g don't exist because the functions, f(x)= g(x)= A- x, where A is any constant, do, in fact, satisfy f(x+ y)= A- x- y= g(x)- y= A- x- y.
     
  4. Nov 6, 2015 #3
    Oh dear, thank you so much for the help HallsofIvy, I had been struggling to prove this for so long, I am happy to have found PF so I could get this off my chest. I encountered this question from one of the older editions of Spivak, I had no idea of this error :oldbiggrin:
     

    Attached Files:

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prove that no such functions exist
  1. Proving existence (Replies: 4)

Loading...