Some recent maths activity

Yesterday I received the question from my engineering friend:

Let $f$ be a real function so that for all $x,y\in \mathbb{R}$, $f(x+y) = f(x)+f(y)+xy(x+y)$ and $\lim _{x \to 0} f(x)/x = 1$ hold. Find $f$.

It does not like a casual question to ordinary university students, not even for maths students...but anyway one may notice that $xy(x+y) = (x+y)^3 - x^3 - y^3$ if you know symmetric polynomials well, and the free linear term makes up the limit we have $f(x) = x^3/3 + x$.

What about uniqueness?

Well, it is easy to show that the function is continuous by exploiting the equality $f(2x) = 2f(x) + 2x^3$, but even stronger we can prove differentiability. Rearranging gives

$\frac{f(x+y)-f(y)}{x} = \frac{f(x) + xy(x+y)}{x}$

Taking limit $x\to 0$ yields $f'(y) = 1 + y^2$ - not only that the derivative exists, we also get a complete DE with an initial value $f(0) = 0$. That easily solves to $f(x) = x^3/3 + x$.

What if the limit condition is changed? Say, $\lim _{x\to 1} f(x)/(x-1) = 1$? We can rewrite the expression as the following:

$f(x+y-1) = f(x-1)+f(y)+(x-1)y(x+y-1)$

Dividing both sides by $(x-1)$ reduces the question to the original case which gives the same solution.

Let is consider the functional equation at a much generalized form: $f(x+y) = f(x)+f(y)+g(x,y)$. According to the above argument if we managed to show that
$\lim _{x\to 0} (f(x)+g(x,y))/x$ exists then we can easily reduce it back to a DE where existance or uniqueness is clear. However this is hard to work around if we do not assume the limit condition because we know pretty much nothing about $f$. It does not work by assuming continuity of $f$ or $g$, since we may come across to some very nasty functions like the Weierstrass function which makes no sense in these questions. We leave a few observations here without solving it (or even getting close):

1. $g$ must by symmetric. This is obvious by observing the rest of the term. In particular, if $g$ is a polynomial then it is in the ring generated by $\sigma _1 = x+y$ and $\sigma _2 = xy$.

2. If $g(x,y) = O(xy)$ for small $x,y$ then it is possible to recover $\lim _{x\to 0}f(x)/x = 1$ using estimates like $f(x) = 2^n f(2^{-n}x) + O(x^2)$ or $f(x) = nf(x/n) + \log n O(x^2)$.

3. If $g$ is Lipschitz we know immediately that it's differentiable a.e. but that says we could have uncountably many non-differentiable points that we not want to deal with...

But that's it. I do not want to spend more than 60 minutes on this useless (for me) problem :d

------------

The 1st Simon-Marais (aka the Pacific Putnam) was held on October 2017 and the statistics are finally out (compare the efficiency against IMO marking team...). It's very surprising that only 1~2% of the students got problem A4 (and A3). I expected the top rankers to be close to 42 (aka 6 correct answers) but it turned out that not many olympiad players participated the event as can be judged from the award list. I expected the event to be much harder next year.

$\