Remark 2

The function $x\mapsto \varrho (x)$ on A, being the greatest lower bound of the continuous functions $x\mapsto \|x^n\|^{1/n}$ for $n\geqslant 1$, is upper semi-continuous (TG, IV, p. 31, cor.), but in general it is not continuous. It may even happen (cf. exerc. 12 of I, p. 157) that a sequence of nilpotent elements of A tends to an element which is not quasi-nilpotent.