Find all continuous functions $f:\mathbb R^{+}\to\mathbb R^{+}$ such that $f(x)^2=f(x^2)$ for all positive reals $x$.
If $f$ is a constant function, then $f(x)=1$. If $f$ is non-constant, I'm suspecting the only solutions are of the form $f(x)=x^k$, where $k$ is a constant, but I have no idea how to prove this. Thanks in advance.
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 $x<1$.