This book provides a rigorous introduction to the techniques and results of real analysis
metric spaces and multivariate differentiation suitable for undergraduate courses. Starting
from the very foundations of analysis it offers a complete first course in real analysis
including topics rarely found in such detail in an undergraduate textbook such as the
construction of non-analytic smooth functions applications of the Euler-Maclaurin formula to
estimates and fractal geometry. Drawing on the author's extensive teaching and research
experience the exposition is guided by carefully chosen examples and counter-examples with
the emphasis placed on the key ideas underlying the theory. Much of the content is informed by
its applicability: Fourier analysis is developed to the point where it can be rigorously
applied to partial differential equations or computation and the theory of metric spaces
includes applications to ordinary differential equations and fractals. Essential Real Analysis
will appeal to students in pure and applied mathematics as well as scientists looking to
acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty
including some suitable for group work or class discussion make this book suitable for
self-study as well as lecture courses.
