How can $f(x,y)$ be written as a function of $g\left(\frac{y}{x}\right)$?

Question

If $f(tx,ty)=f(x,y)$, how does it imply that: $$f(x,y)=g\left(\dfrac {y}{x}\right)$$

Yesterday I asked a question on ordinary differential equations and got an answer. The answerer used the above stated "theorem" but I didn't understand why it is always true. Also why is $t$ (which is a constant) set to $1/x?$ Why can't we set $t=1/(x^2+1)$ or anything else? Is it that we can set $t$ to any function of $x$ that makes my life easy? Why is it so?

Answer 1

For any fixed $(x,y)\neq (0,0)$, the set $\{(tx,ty), t\in\mathbb{R}\}$ represents the whole line through the origin in the direction of $(x,y)$. The condition $f(tx,ty)=f(x,y)$ means that $f$ is constant along such line and thus, it only depends on its slope $y/x$. These functions are called homogeneous (more exactly, homogeneous of order zero). Since they are constant along lines through the origin, they are discontinuous at the origin unless they are constant on the whole space. In the context of differential equations, you introduce a new variable $u=y/x$ which is precisely the slope of the line to reduce to a separable equation.

Answer 2

Hint: it is easy. first set $t$ to $\frac{1}{x}$. You will get $f(1,\frac{y}{x}) = f(x,y)$.