Turtles all the way

Let $g(x) = \frac{1}{x+\frac{1}{x+\frac{1}{...}}} \Rightarrow g = \frac{1}{x+g} \Leftrightarrow g^2 + xg - 1 = 0$ . So:

$g_1(x) = \frac{-x+\sqrt{x^2 +4}}{2}$ and $g_2(x) = \frac{-x-\sqrt{x^2 +4}}{2}$

We now need to determine if any of these solutions is an extraneous solution. In particular, (as pointed out by Leonel,) since $g(x)$ must send positive values of $x$ to positive values of $g(x)$ , so we have that $g(x) = \frac{-x+\sqrt{x^2 +4}}{2} = f'(x)$ , now we can just solve it numericaly to get the answer. Here is my code for the numeric solution and the graph of $f(x)$ :

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import matplotlib.pyplot as plt
import numpy as np

def g(x):
    return (-x + np.sqrt(x**2 +4))/2

def f(x, x0, f0):
    n =  [x0]
    fn = [f0]

    step_size = 0.00001
    while not np.allclose(x, n[-1]):
        fn.append(fn[-1] + step_size*g(n[-1]))
        n.append(n[-1] + step_size)

    plt.plot(n, fn, label="f(x)")
    plt.legend()
    plt.show()
    return fn[-1]

print("%.2f" % f(3, 0, 0))