Example 6

Let $f(x, y)$ be a polynomial with real coefficients in two real variables $x, y$, not having a constant term. If, as $x$ approaches 0 while remaining $> 0$, there is a function $\varphi(x)$, continuous on an interval $[0, a]$, and such that $\varphi(0) = 0$ and $f(x, \varphi(x)) = 0$ for $0 \leq x \leq a$, one can show that there are a rational number $r$ and a real number $\lambda \neq 0$ such that $\varphi(x) \sim \lambda x^r$ (V, p. 259, exerc. 3).