THIS IS THE SECOND PART OF THE AUTHOR'S TEXT SERIES ON REAL ANALYSIS, INTENDED TO BE A BASIC TEXT FOR THE UNDERGRADUATE HONOURS COURSE IN MATHEMATICS OF THE INDIAN UNIVERSITIES.  ALL THE NECESSARY TOPICS ON ELEMENTARY REAL ANALYSIS ARE COVERED IN THIS BOOK IN A SIMPLE YET SYSTEMATIC MANNER. THE TREATISE CONTAINS SOME VERY USEFUL AND DISCRETE TOPICS LIKE COMPACTNESS, UNIFORM CONVERGENCE OF SEQUENCES AND SERIES OF FUNCTIONS, POWER SERIES, FUNCTIONS OF BOUNDED VARIATION, TAYLOR'S THEOREM FOR FUNCTIONS OF TWO VARIABLES, EXTREMES(BOTH FREE AND CONSTRAINED) OF FUNCTIONS OF SEVERAL VARIABLES, DOUBLE SEQUENCES AND DOUBLE SERIES. BESIDES THESE CLASSICAL TOPICS, TWO IMPORTANT SECTIONS HAVE ALSO BEEN APPENDED- ONE IS ON METRIC SPACES AND OTHER IS ON ELEMENTS OF COMPLEX ANALYSIS.  THESE TWO TOPICS HAVE BEEN RECENTLY INCLUDED  IN SOME OF THE UNIVERSITIES (INCLUDING THE UNIVERSITY OF CALCUTTA) IN THE UNDER GRADUATE HONOUS COURSES AS PART OF IMPLEMENTATION PROGRAMME OF IMPROVEMENT OF UNDERGRADUATE SYLLABI SUGGESTED BY THE UNIVERSITY GRANTS COMMISSION.