  We systematically derive general properties of continuous and holomorphic
functions with values in closed operators, allowing in particular for operators
with empty resolvent set. We provide criteria for a given operator-valued
function to be continuous or holomorphic. This includes sufficient conditions
for the sum and product of operator-valued holomorphic functions to be
holomorphic.
  Using graphs of operators, operator-valued functions are identified with
functions with values in subspaces of a Banach space. A special role is thus
played by projections onto closed subspaces of a Banach space, which depend
holomorphically on a parameter.
