Question 187
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous, bounded and non-constant function satisfying
$$
f(2x) = 2f(x)^2 - 1
$$
for all $x \in \mathbb{R}$. Show that the equation $f(x) = 0$ has a solution.
$$
f(2x) = 2f(x)^2 - 1
$$
for all $x \in \mathbb{R}$. Show that the equation $f(x) = 0$ has a solution.