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This is a mathematical reference book for students, graduates and researchers who are interested in the challenging problems as the extreme of function, equation and series arising in the mathematical analysis. The book aims at encouraging those who were accustomed to standard calculation to be interested in solution of more challenging problems such as differential equation, integral equation and series. This research paper is the limit part of "challenging problems and elegant solution in mathematical analysis" to continuity of limit part, integral part and equation part. It can be widely used by students in mathematics and physics, and everyone who is interested in mathematical analysis. It is certain that this will be contributed to the students, the graduates and researchers who are going to mathematically analyze the physical situations arise in advanced science and technology.