In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${lambda}{in}{sigma}_{s-F}(T){setminus}overline{{ ho}^{pm}_{s-F}(T)}$ and ${in}$ > 0, the ball $B({lambda},{in})$ contains a component of ${sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.