It even has a proof of Weierstrass' approximation theorem

The eigth chapter introduces familiar functions in a new light, such as the gamma and trigonometric functions

Also, after a quick introduction of the real and complex number fields, Rudin quickly presents enough topological concepts to truly understand analysis on the real line, and more generally, finite-dimensional Euclidean space

On the down side, the latter sections (chapter nine onward) on differential forms and introductory Lebesgue measure leave much to be desired

Sure it was OK before Rudin wrote the thing, but now? Why spit on your luck? And if you'r a student and find the book too hard? Try harder

While this is not a building block to the Lebesgue theory, it is useful because it shows that the Riemann integral is merely an introduction to integration that works in only nice cases

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