Prove that no such functions exist

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1. Nov 6, 2015

lordianed

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. Nov 6, 2015

HallsofIvy

Staff Emeritus
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.

3. Nov 6, 2015

lordianed

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

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