Find all functions $f$ such that $f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)$.

Question

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)\text.$$

I have derived these clues:

• $f(0)=0$;
• $f(x^2)=xf(x)$;
• $f(x)=-f(-x)$.

But now I am confused. I know solution will be $f(x)=x$, but I don't know how to prove this.

Answer 1

$f(x)+f(x+1)=f(2x+1)=(2x+1)(f(x+1)+f(-x))$.

That can be rearranged to

$$\frac{f(x)}{x}=\frac{f(x+1)}{x+1}$$

so, for example, $f(n)=nf(1)$

Answer 2

I don't know how to do this without continuity. If you can show $f(x) \to f(0)$ whenever $x \to 0$, then the original expression yields it. Then all you need is $f(x)$ bounded near zero in order to use $f(x^2) = x f(x)$ to get that the limit exists and is zero.

However, taking continuity for granted, consider your result $f(x^2)=xf(x)$ which can be rewritten as $f(x) = x^{1/2} f(x^{1/2})$ if we let $x^2 \to x$ for positive $x$. Then $f(x^2) = x \cdot x^{1/2} f(x^{1/2})$. But then we can write $f(x^{1/2}) = x^{1/4} f(x^{1/4})$ if we let $x \to x^{1/4}$ in $f(x^2) = xf(x)$ so that $f(x^2) = x \cdot x^{1/2} \cdot x^{1/4} f(x^{1/4})$.

Repeating this $n$ times we get $$f(x^2) = x^{\sum_{n=0} 2^{-n}} f(x^{2^{-n}}) = x^{(2-2^{-n})} f(x^{2^{-n}})$$ which holds for $x>0$. Now for $x > 0$, taking the limit as $n \to \infty$ gives $$f(x^2) = x^2 f(1)$$ using continuity of $f$ or restated $$f(x) = f(1) x$$ for positive $x$.

But now for negative $x$ use the fact $f(x) = -f(-x)$ so that $f(-x) = -x f(1)$. Letting $\alpha = f(1)$, we say all solutions must be of the form $f(x) = \alpha x$. We easily verify they are all solutions.

Answer 3

Let $P(x, y)$ denote $$f(x^2-y^2)=(x-y)(f(x)+f(y))$$

$P(0, 0)$ gives $f(0)=0$, then $P(x, 0)$ gives $f(x^2)=xf(x)$.

Now $-xf(-x)=f((-x)^2)=f(x^2)=xf(x)$. Thus for $x \not =0$, $f(-x)=-f(x)$.

This clearly holds for $x=0$ as well, so $f(-x)=-f(x)$.

$P(x, -y)$ gives $$f(x^2-y^2)=(x+y)(f(x)+f(-y))=(x+y)(f(x)-f(y))$$

Comparing with $P(x, y)$ gives $$(x+y)(f(x)-f(y))=(x-y)(f(x)+f(y))$$

Thus $2yf(x)=2xf(y)$. Putting $y=1$ gives $f(x)=xf(1)$. We may easily verify that these are indeed solutions.

Answer 4

I have found a proof of $f(x)=xf(1)$ without assuming continuity, but only for $x$ integer.

First, let $u=y+x$, $v=y-x$. Then $$f(uv)=v\left[f\left(\frac{u+v}2\right)+f\left(\frac{u-v}2\right)\right]$$

Now let $u=1$, $v=t$ to get $$f(t)=t\left[f\left(\frac{t+1}2\right)-f\left(\frac{t-1}2\right)\right]$$ and $u=t$, $v=1$ yields $$f(t)=\left[f\left(\frac{t+1}2\right)+f\left(\frac{t-1}2\right)\right]$$ Combining these equations it's not difficult to obtain $$f\left(\frac{t+1}2\right)=f\left(\frac{t-1}2\right)\frac{t+1}{t-1}$$ For $t$ an odd number $\geq3$, write $n=(t-1)/2$ and $$f(n)=f(n-1)\frac{n}{n-1}$$ Apply recursively this formula and $$f(n)=nf(1)$$