# 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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