This English version of the clear, rigorous and complete Italian textbook of Mathematical Analysis is designed to back beginners' classes in Mathematics in all curricula of scientific nature where mathematical tools are a significant part of the education. It reflects a two-decade teaching practice during which it was employed with major returns. The criterion adopted is to present the subject matter in a clear and immediately applicable way, without renouncing the exposition's rigour or reducing to a mere collection of recipes and formulas. The fundamental concepts and techniques regarding limits, continuity, differential and integral calculus for functions of one real variable are presented with the primary intention to train students to use them effectually and at the same time critically.