Reference request: continuity of Cholesky factor

Numerical Analysis: A Mathematical Introduction page 295.

A subtle issue is that $\Pi$ is not unique here. For instance, if

$$ A = \begin{bmatrix} 1 & 0 & 0\\\\ 0 & 0 & 0\\\\ 0 & 0 & 0 \end{bmatrix} $$

then you can take both the identity and $(23)$ as the permutation. Similarly, if $A=I$, then any $\Pi$ will work (and $R=I$).

I don't think you can speak about continuity until you resolve this ambiguity.