  This paper is concerned with dependence of discrete Sturm-Liouville
eigenvalues on problems. Topologies and geometric structures on various spaces
of such problems are firstly introduced. Then, relationships between the
analytic and geometric multiplicities of an eigenvalue are discussed. It is
shown that all problems sufficiently close to a given problem have eigenvalues
near each eigenvalue of the given problem. So, all the simple eigenvalues live
in so-called continuous simple eigenvalue branches over the space of problems,
and all the eigenvalues live in continuous eigenvalue branches over the space
of self-adjoint problems. The analyticity, differentiability and monotonicity
of continuous eigenvalue branches are further studied.
