I'm reading the yellow book (CFT by Di Francesco) and I came across a bit of a confusing statement.
Consider a scaling transformation $x^\mu \rightarrow x'^\mu = \lambda x^\mu$. Then the metric components transform as $g'_{\mu\nu}(x') = \frac{g_{\mu\nu}}{\lambda^2}$ and hence $x'_\mu = \frac{x_\mu}{\lambda}$. Now in (4.50) the author claims that by rotations and translations we can constrain the 2point function $$\langle \phi(x_1) \phi(x_2) \rangle = f(|x_1-x_2|).$$ Since it must hold that $$\langle \phi(x_1) \phi(x_2) \rangle = \lambda^{2\Delta}\langle \phi(\lambda x_1) \phi(\lambda x_2) \rangle,$$ we must have that $$f(x) = \lambda^{2\Delta}f(\lambda x).$$ However, since we fixed $f$ to only depend on the distance between $x_1$ and $x_2$, it should be invariant under a scaling, since $$x^2 = x^\mu x_\mu \rightarrow x'^\mu x'_\mu = \lambda x^\mu \frac{x_\mu}{\lambda} = x^2.$$
What am I misunderstanding here?
EDIT: I think my question boils down to the following: If $x'^\mu =\lambda x^\mu$ and $g'_{\mu\nu}(x') = \frac{g_{\mu\nu}}{\lambda^2}$, then how does $x_\mu$ transform here? This is not discussed in the linked question.
