In　this　paper,　we　provide　some　necessary　and/or　sufficient　conditions　of　(linear　or　nonlinear)　error　bounds　for　proper　functions　(not　necessarily　lower　semicontinuous).　As　applications,　we　present　some　linear　and　nonlinear　metric　regularity　and　subregularity　results　for　the　set-valued　mappings　with　closed-graph.