This book contains the English version of a one-semester undergraduate course through the medium of Irish. I have given this course in Maynooth each year since 2006. The class is typically quite mixed, and the book aims to challenge students familiar with rigorous arguments while simultaneously providing comfort to other students that prefer procedural mathematics---not an easy balancing act! For instance, some exercises are targeted at one or other of these two groups of students and, while some material is done in the context of metric spaces, guidance is given to the reader who prefers to stick to the real line. This second edition of the book has no major new topics, but it is about twice as long as the first. Some topics are explored in greater depth (e.g. conjugacy and semiconjugacy, bifurcation) and some proofs have been added (notably for Sharkovsky's and Singer's theorems). Additional exercises have been given, and the most challenging ones are marked appropriately.