Proof of first Bianchi identity

Question

The proof is often simplified by using the following theorem:

"If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{ij}/\partial y^k,\partial g^{ij}/\partial y^k, \Gamma_{ijk}$, and $\Gamma^i_{jk} $ are zero."

It follows that at $O$, the origin of normal coordinates: $$R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}$$ (1.0)

The whole proof will not be shown here.

I understand that the above expression comes directly from the definition of the Riemann Tensor of the first kind $R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}+\Gamma_{ilr}\Gamma^{r}_{jk}-\Gamma_{ikj}\Gamma^r_{jl}$ but if all the Christoffel symbols vanish at the origin of the Riemannian(normal) coordinates then why do the first two terms (i.e. the RHS of (1.0)) remain?

Answer 1

For the same reason that given a function $f(x)$ which zero at some $x=x_0$, it does not imply that $f'(x_0)=0$.