The map which takes a bounded operator to its spectrum, is upper semicontinuous, but its continuity points form a proper dense. Two problems thus arise:
- at which points of B(H) is F in fact continuous
- what continuity-type properties hold.
Requirements: Functional analysis, subharmonic functions and possibly several complex variables. References:  B. Aupetit, A primer on spectral theory, Springer-Verlag, New York, 1991.  J.B. Conway & B.B. Morrel, Operators that are points of spectral continuity, Integral Equations and Operator Theory 2 (1979), 174-198.