This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first-year students who study mathematics.
The present edition covers essentially the same topics as the second one,with some additions,a few minor omissions,and considerable rearragement.I hope that these changes will make the material more accessile and more attractive to the students who take such a course.
Experience has convinced me that it is pedagogically unsound(though logically correct)to start off with the construction of the real numbers from the rational ones.At the beginning,most students simply fail to appreciate the need for doing this.Accordingly,the real number system is introduced as an ordered field with the least upper-bound property,and a few interesting applications of the property are quickly made.However,Dedekind's Construction is not omitted.It is now in an Appendix to Chapter 1,where it may be studied and enjoyed whenever it seems ripe.
The material of funcions of several variables is almost completely rewritten,with more details fill in,and with more examples and more motivation.The proof of the inverse function theorem--the key item in Chapter 9--is simplified by means of the fixed point theorem about constractionh mappings.Differential forms are discussed in much greater detail.Several applications of Stokes' theorem are included.
As regard as other changes,the chapter on the Riemann-Stieltjes integral has been trimmed a bit,a short do-it-yourself sectioin on the gamma funcion has been added to Chapter 8,and there is a larger number of new exercises,most of them with fairly detailed hints.
I have also included several references of articles appearing in the American Mathematical Monthly and int Mathematics Magazine,in the hope that student will develop the habit of looking into the journal literature.Most of these refrences were kindly supplied by R.B.Burckel.
Over the years,many people,student as well as teachers,have sent me corrections,critisms,and other comments concerning the previous editions of the book.I have appreciated these,and I take this opportunity to express my sincere thanks to all who have written me.
WALTER RUDIN

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The Home
Rabindranath Tagore
I paced alone on the road across the field while the sunset was hiding its last gold like a miser.
The daylight sank deeper and deeper into the darkness,and the widowed land,whose harvest had been reaped,lay silent.
Suddently a boy's shrill voice rose into the sky.He traversed the dark unseen,leaving the track of his song across the hush of the evening.
His village home lay there at the end of the waste land,beyond the sugar-cane field,hidden among the shadows of the banana and the slender areca palm,the cocoa-nut and the dark green jack-fruil trees.
I stopped for a moment in my lonely way under the starlight,and saw spread before me the darkened earth surrounding with her arms countless homes furnished with cradles and beds,mothers' hearts and evening lamps,and young lives glad with a gladness that the knows nothing of its value for the world.

On The Seashore
Rabindranth Tagore
On the seashore of endless worlds children meeting.
The infinite sky is motionless overhead and the restless water is boisterous.On the seashore of endless world the children meeting with shouts and dances.
They build their houses with sand,and they play with empty shells.With withered leaves the weave their boats and smilingly float them on the vast deep.Children have their play on the seashore of the worlds.
They know not how to swim,they know not how to cast nets.Pearl-fishers dive for pearls,merchants sail in their ships,while children gather pebbles and scatter them again.The seek not for the hidden treasures,the know not how to cast nets.
The sea surges up with laughter,and pale gleams the smile of the sea-beach.Death-dealing waves sing meaningless ballads to the children,even like a mother while rocking her baby's cradle.The sea plays with children,and pale gleams the smile of the sea-beach.
On the seashore of endless worlds children meet.Tempest roam in the pathless sky,ships are wrecked in the trackless water,death is abroad and children play.On the seashore of endless worlds is the greatmeeting of children.