Find all continuous functions from positive reals to positive reals such that $f(x)^2=f(x^2)$
Why, the solutions are plenty, even the continuous ones. Let $f(2)=a$; then $f(4)=a^2$. Now draw a freehand curve from the point $(2,a)$ to $(4,a^2)$. That would be your function. Using $f(x^2)=f(x)^2$, extend it to $[4,16]$, then to $[16,256]$, and so on. Then use the equation in reverse and extend to $[\sqrt2,2]$ and so on. Then maybe we'll have to repeat the entire trick for $x<1$.