A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

Solution 1:

Such a function doesn't exist.

Forget $f(1) = 1$. Just assume $f(x)$ satisfy the functional equation on $[0,\infty)$. We have

$$ \begin{align} f(x) &= 1 + x f\left(\frac{1}{x+1}\right)\\ &= 1 + x \left[ 1 + \frac{1}{x+1}f\left(\frac{x+1}{x+2}\right)\right]\\ &= 1 + x \left[ 1 + \frac{1}{x+1} + \frac{1}{x+2} f\left(\frac{x+2}{2x+3}\right) \right]\\ &\;\vdots\\ &= 1 + x \left[ 1 + \left( \sum_{k=2}^{p-1} \frac{1}{F_{k-1} x + F_k} \right) + \frac{1}{F_{p-1} x + F_p}f\left(\frac{F_{p-1} x + F_p}{F_p x + F_{p+1}}\right)\right]\\ &\;\vdots \end{align} $$ where $F_k$ is the $k^{th}$ Fibonacci number. Notice

$$\lim_{p\to\infty} \frac{F_{p+1}}{F_p} = \phi = \frac{1+\sqrt{5}}{2} \quad\implies\quad \lim_{p\to\infty} \frac{F_{p-1} x + F_p}{F_p x + F_{p+1}} = \frac{1}{\phi} $$

If we further assume $f(x)$ is continuous at a single point $\displaystyle\;\frac{1}{\phi}$, the last expression converges as $p \to \infty$ and we get:

$$f(x) = 1 + x + \sum_{k=2}^\infty \frac{x}{F_{k-1} x + F_k}$$ Substitute $x$ by $1$, we get

$$f(1) = 1 + 1 + \sum_{k=2}^\infty\frac{1}{F_{k-1}+F_k} = \frac{1}{F_1} + \frac{1}{F_2} + \sum_{k=3}^\infty \frac{1}{F_k} = \sum_{k=1}^\infty \frac{1}{F_k}$$

The number at the RHS is known as the Reciprocal Fibonacci constant $\psi$ with a value $$\approx 3.359885666243177553172011302918927179688905133731$$ differs from $1$.

Solution 2:

There is no such function. Let $x_0=1$, $x_{n+1}=\frac1{x_n+1}$. Note that all $x_n\in(0,1]$. Now, consider $$ f(x_{n+1}) = \frac{f(x_n)-1}{x_n} $$ as a recurrence for the sequence $f(x_n)$. We compute that $f(x_2) = -2 < 0$. Since all $x_n > 0$, this implies (by induction, say) that $f(x_n)<0$ for $n\ge 2$. But this means the numerator of the RHS is negative; thus the fact that $x_n\le 1$ implies $$ f(x_{n+1}) \le f(x_n)-1 $$ and so $f(x_n)\to -\infty$, violating continuity. (I'm taking for granted here that $x_n$ converges; indeed, $x_n$ is the sequence of convergents of $\frac{\sqrt5-1}{2}$.)

Solution 3:

This is not a full solution, but instead I wanted to give two thoughts that are too long for the comments section.

1) Note that since $f(0) = 1$ and $f(1) = 1$ you can rearrange your expression to $$ \frac{f(x) - f(0)}{x-0} = f \Big( \frac{1}{x+1} \Big) $$ for any $x \neq 0$. Taking a limit on both sides and using the continuity of $f$, you find that $f'(0) = f(1) = 1$. So, at very least, any function satisfying your criteria will be differentiable at $0$. In particular, if a function exists which is smooth enough that you can perform a Taylor expansion near $0$, you will have $f(x) = 1 + x + O(x^2)$.

2) If you can find $f(x)$ for all $x \in (0,1)$ you'll be done since for any $y \in (1, \infty)$ you have $\frac{1}{1+y} \in (0,\frac{1}{2})$ (here you can see $(0,1)$ is even more than you actually need, you really only need $(0,\frac{\sqrt{5}-1}{2}]$, but the idea is the same). So, if you are able to use your reasoning above to find $f(q)$ for every $q \in \Bbb{Q} \cap (0,1)$, then using continuity and density arguments, you can find $f(x)$ for every $x \in (0,1)$ and then be done.

Hopefully this will help.