Function Composition

Are there real-valued continuous functions $f,g$ defined on real numbers such that
$$f(g(x)) = x^{2011}$$
and
$$g(f(x)) = 2011x$$

for all $x$?

Solution

Applying $g$ to both sides, we have:
$$g(f(g(x))) = g(x^{2011})$$
$$2011g(x) = g(x^{2011})$$

Let $x = 1$, we have $g(1) = 0$

Let $P(y) = g(e^y)$ defined on all $y$ real numbers.
So $P(2011y) = g(e^{2011y}) = g((e^y)^{2011}) = 2011g(e^y) = 2011P(y)$
We conjecture that $P(y) = ay$ for some $a$. Setting $x = e^y$, we have:
$$g(x) = g(e^y) = P(y) = ay = a \log x$$

Now plugging back in, our second equation becomes:
$$a \log f(x) = 2011 x$$
$$f(x) = e^{2011x / a}$$

Testing for compatibility with first equation:

$$f(g(x)) = e^{2011 a \log x / a} = e^{2011 \log x} = x^{2011}$$