Let $x \in \mathbb{R}$, $x \notin \mathbb{Q}$ and let the function $f:\mathbb{R}\setminus \mathbb{Q} \,\times \mathbb{N}\rightarrow \mathbb{Q}$ provide the best rational approximation for $x$ where the denominator is less than or equal to a given number $N \in \mathbb{N}$. E.g. $f(\pi, 50) = \frac{22}{7}$.
Let $f(x, N) = \frac{a}{b} > 0$, is it possible to prove that $f(\frac{1}{x}, a) = \frac{b}{a}$? Is it even perhaps possible to identify $M > a$ and make the stronger statement that $f(\frac{1}{x}, M) = \frac{b}{a}$?
Would appreciate suggestions on a good approach to take.