Analysis I (part 4) : Limit of a function, continuous functions

We introduce real functions of one real variable. We begin by defining some of their properties, including monotonicity, parity and periodicity, as well as operations between functions. We define special functions such as hyperbolic functions. We continue our study of functions, defining step-defined functions, in particular Signum and Heaviside functions. Important practical manipulations on functions are affine transformations. We finally get to the heart of the matter by defining the pointed limit of a function at a point, and give examples of function limits. We end this discussion with the concept of left and right limits. In what follows, we return to the study of the limit of a function, starting by defining algebraic operations on limits. We then study the infinite limits of functions. In order to be able to calculate the limits of functions, we give the Two Gendarmes Theorem and discuss some examples with algebraic, exponential and trigonometric functions. We take up the concept of the blunt limit defined earlier, giving a different but equivalent definition. We introduce the concept of continuity. We define it in two different ways, as for the limits of functions. Finally, we use continuity to extend certain functions, and study continuity on open intervals.