All 2D differentiable surfaces conformal to the euclidian metric?

Question

Is it true that the metric of any continuous differentiable 2D surface is locally conformal to the euclidian metric ? (see equ. (2) below). Can I state something like this in general ? If so, what is the name of the theorem that gives a demonstration ? If it's not general, what are the conditions to make this true ? \begin{align}\tag{1} ds^2 = g_{ij} \, dx^i \, dx^j &\equiv g_{11} \, dx^2 + g_{22} \, dy^2 + 2 \, g_{12} \, dx \, dy \\[12pt] &= \Omega^2(u, v)(du^2 + dv^2). \tag{2} \end{align} The original metric (in coordinates $x$, $y$) is positive definite.