MathOverflow

Let $f:(0,\infty) \to \mathbb{R}$ be a function satisfying $$ \frac{f(x+y) - f(y)}{x} = \frac{f(x+y) - f(x)}{y} $$ for all $x,y>0$.Does this already imply that $f$ is an affine function, i.e., $f(x) = ax + b$ for some $a,b \in \mathbb{R}$?
I was able to show that this holds under the additional assumption that $\lim_{x \searrow 0} f(x)$ exists, but I am not sure whether this assumption is really necessary....

Let $K$ be a number field. Let $L$ be the normal closure of $K$.
Since $L$ is Galois, it has many automorphisms. If the Galois group has many cosets, there are potentially many automorphisms that can map fundamental units of L to other (independent) units of L....

Background. A friend and I who work in the representation theory of $p$-adic groups are trying, for culture, to understand some basic ideas from the relative Langlands program of Sakellaridis and Venkatesh. Without going too much into the details of...

Suppose $k$ is an algebraically closed field in characteristic $0$ (say, $k \cong \overline{\mathbb{Q}}$), and let $\mathrm{PG}(m,k)$ be the $m$-dimensional projective space over $k$ (for some fixed positive integer $m$). Suppose one can prove the following...