Thisbookisfirstofalldesignedasatextforthecourseusuallycalled"theoryoffunctionsofarealvariable".ThiscourseisatpresentcustomarilyofferedasafirstorsecondyeargraduatecourseinUnitedStatesuniversities，althoughtherearesignsthatthissortofanalysiswillsoonpenetrateupperdivisionundergraduatecurricula.Wehaveincludedeverytopicthatwethinkessentialforthetrainingofanalysts，andwehavealsogonedownanumberofinterestingbypaths.Wehopetoothatthebookwillbeusefulasareferenceformaturemathematiciansandotherscientificworkers.Hencewehavepresentedverygeneralandcompleteversionsofanumberofimportanttheoremsandconstructions.Sincethesesophisticatedversionsmaybedifficultforthebeginner，weavegivenelementaryavatarsofallimportanttheorems，withappropriatesuggestionsforskipping.Wehavegivencompletedefinitions，explanations，andproofsthroughout，sothatthebookshouldbeusableforindividualstudyaswellasforacoursetext