this is how Carroll introduces the concept of locally inertial coordinates:
It is always possible to put the metric into canonical form. In fact it is always possible to do so at some point $p\in M$, but in general it will only be possible at a single point, not in any neighborhood of p. Actually we can do slighlt better than this; it turns out that at any point p there exists a coordinate system $x^{\mu}$ in which $g_{\mu\nu}$ takes its canonical form and the first derivatives $\partial_{\sigma}g_{\mu\nu}$ all vanish: