Functions satisfying $f(m+f(n)) = f(m) + n$

I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you help me out by solving, proving and explaining (it would be nice if you could) the following question? Thanks, any help is appreciated.

I think one of my teachers mentioned something about plugging in $m=0$ or something, but... Bleh, I don't know. Please help?

This question is taken from the South African Mathematics Olympiad of 1997.

Find all functions $f: \Bbb Z \to \Bbb Z $ which satisfy

$f(m+f(n)) = f(m) + n$

for all $m$, $n$ which are integers.

Thanks :)

Thanks to everyone who helped me with this question :) I'm now one step further in my life...

Plugging in $n=0$, we get that $$f(m+f(0))=f(m)\tag{1}$$ must hold for all $m\in\mathbb Z$. There are now two possibilities.

Case 1. $f(0)\neq 0$.

In this case, the equation $(1)$ tells us that $f$ is a periodic function. But this would necessarily imply that $f$ is bounded (since it would have to take only finitely many values), i.e. there exists a positive integer $M$ such that $-M\leq f(x)\leq M$ holds for all $x\in\mathbb Z$. But this is impossible: plugging in $n = 2M+1$ into the equation would then yield $$2M\geq f(m+f(2M+1))-f(m)=2M+1$$ which is absurd; therefore this case does not occur.

Case 2. $f(0) = 0$.

In this case, plug $m=0$ into the original equation. This tells us that $$f(f(n))=n\tag{2}$$ must hold for all $n\in\mathbb Z$. Plugging in $f(n)$ instead of $n$ into the original equation and using $(2)$ then tells us that $$f(m+n)=f(m)+f(n)$$ holds for all $m,n\in\mathbb Z$. Using the usual argument, this implies that $f$ is of the form $f(x) = c x$ for some constant $c$. By $(2)$, this constant is either $1$ or $-1$. Both choices satisfy the original equation.

Conclusion. The solutions to the equation are given precisely by $f(x)=x$ and $f(x) =-x$.

Clearly, $f(n) = \pm n$ is a solution. Now let us see if there are other solutions. Plug in $n=0$. We then have $$f(m+f(0)) = f(m)$$ If $f(0) \neq 0$, we then have $f$ to be periodic with period $f(0)$. This means $f(n)$ an take only finitely many values. Plugging $m=0$, we get that $$f(f(n)) = n + f(0)$$ which takes infinitely many values contradicting the previous fact. Hence, $f(0) = 0$. Now plug in $m=0$ to get $$f(f(n)) = n$$ Hence, we have $$f(m+n) = f(m+f(f(n))) = f(m) + f(n)$$ This means $$f(m) = \text{constant} \times m = km$$ This means $$f(m+f(n)) = f(m) + n \implies f(m+kn) = km+n \implies km + k^2n = km +n \implies k^2n = n$$ Since this is true for all $n$, we have $k^2=1$. Hence, $k = \pm 1$. Hence, the only solutions are $$f(n) = \pm n$$

Let the given assertion be $P(m,n)$ $$P(0,n) \implies f(f(n))=n+f(0)$$ So, let's assume $f(a)=f(b)$ $$f(a)=f(b)\implies f(f(a))=f(f(b)) \Leftrightarrow a+f(0)=b+f(0) \Leftrightarrow a=b$$ That means that $f$ is injective (one-to-one), so $f(a)=f(b)\Leftrightarrow a=b$ Now, $$P(0,0)\implies f(f(0))=f(0) \Leftrightarrow f(0)=0$$ So, (by $P(0,n)$) $$f(f(n))=n$$ Now, $$P(m,f(n))\implies f(m+n)=f(m)+f(n)$$ This is a well-known functional equation called the Cauchy functional equation.

One way of dealing with it: $$\underbrace{f(n)+f(n)+\dots+f(n)}_\text{$k$ times}=f(\underbrace{n+n+\dots+n}_\text{k times})$$ $$\Leftrightarrow kf(n)=f(kn)$$ and putting $n=1$ we get $f(k)=kf(1)$ for all $k \geq 1$

Another way of dealing with it:

Let the conclusion above be $H(m,n)$ $$H(1,1)\implies f(2)=2f(1)$$ $$H(2,1)\implies f(3)=f(2)+f(1)=2f(1)+f(1)=3f(1)$$ $$H(3,1)\implies f(4)=f(3)+f(1)=3f(1)+f(1)=4f(1)$$ and so on, by simple induction, we can say that

$f(x)=xf(1)$ for all $x \geq 1$ and $f(0)=0$

So, for both ways, we need to extend it over the negative integers, by proving that $f$ is odd. Well, $$H(m,-m) \implies f(m)+f(-m)=0 \Leftrightarrow f(-m)=-f(m)$$ Or (another way) $$P(-f(n),n)\implies f(-f(n))+n=0 \Leftrightarrow f(-f(n))=-n$$ $$P(0,-f(n)) \implies f(f(-f(n)))=-f(n) \Leftrightarrow f(-n)=-f(n)$$ So, $f$ is odd, meaning that $f(x)=xf(1)$ applies for all integers $x$

Let $f(1)=c$ giving $f(x)=cx$, substituting in the original equation, we get $$cm+{c^2}n=cm+n$$ $$\Leftrightarrow c^2=1 \implies c=\pm 1$$ So, $$f(x)= \pm x$$