Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f\left(x-f\left(y\right)\right)=1-x-y$, $x,\ y\in\mathbb{R}$

Question

I'm new in functional equations and stuck in this easy problem. Could anyone help with a clear solution?

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f\left(x-f\left(y\right)\right)=1-x-y$, $x,\ y\in\mathbb{R}$

This is what I have done so far:

Let $y=0$, then
$f\left(x-f\left(y\right)\right)=1-x-y\Rightarrow f\left(x\right)=1-x$
$f\left(x-f\left(y\right)\right)=1-x-y\Rightarrow f\left(x-\left(1-y\right)\right)=1-x-y$
$\Rightarrow f\left(x-1+y\right)=1-x-y\Rightarrow 1-\left(x-1+y\right)=1-x-y$
$\Rightarrow 1-x+1-y=1-x-y\Rightarrow 2=1$

But it isn't very helpful.

Thanks, Steve

Answer 1

Your argument only shows that $f(0)\ne0$, but the idea is good if used correctly.

Set $c=f(0)$. Then you know that $$ f(x-c)=1-x $$ for every $x$. For $x=y+c$, you obtain $$ f(y)=1-y-c $$ and the condition $f(0)=c$ implies $c=1-c$.