Analysis forms an essential basis of mathematics as a whole, as well as of the natural
sciences, and more and more of the social sciences too. The theory of analysis (differentiation
and integration) was created—after Galileo’s insight—for the purposes
of describing the universe in the language of mathematics. Working out the precise
theory took almost 300 years, with a large portion of this time devoted to de?nitions
that encapsulate the essence of limits and continuity. Mastering these concepts can
be a dif?cult process; this is one of the reasons why analysis is only barely present
in most high-school curricula.
At the same time, in postsecondary education where mathematics is part of the
program—including various branches of science and mathematics—analysis appears
as a basic requirement. Our book is intended to be an introductory analysis
textbook; we believe it would be useful in any areas where analysis is a part of the
curriculum, in addition to the above, also in teacher education, engineering, or even
some high schools. In writing this book, we used the experience we gained from our
many decades of lectures at the Eotvos Lorand University, Budapest, Hungary.
We have placed strong emphasis on discussing the foundations of analysis: before
we begin the actual topic of analysis, we summarize all that the theory builds upon
(basics of logic, sets, real numbers), even though some of these concepts might be
familiar from previous studies. We believe that a strong basis is needed not only for
those who wish to master higher levels of analysis, but for everyone who wants to
apply it, and especially for those who wish to teach analysis at any level.
The central concepts of analysis are limits, continuity, the derivative, and the
integral. Our primary goal was to develop the precise concepts gradually, building
on intuition and using many examples. We introduce and discuss applications of our
topics as much as possible, while ensuring that understanding and mastering of this
dif?cult material is advanced. This, among other reasons, is why we avoided a more
abstract or general (topological or multiple variable) discussion.
We would like to emphasize that the—classical, mostly more than 100 year old—
results discussed here still inspire active research in different areas. Due to the nature
of this book, we cannot delve into this; we only mention a small handful of unsolved
problems.