Analysis I (part 5): Continuous and derivable functions, the derivative function

This brings us to the heart of our discussion of functions: the concept of the derivability of a function. We are particularly interested in the question of the continuity of derived functions. We begin the chapter by completing the study of continuous functions by studying their properties on closed intervals. This enables us to define the maximum and minimum of continuous functions. We go on to define the bisection method, and demonstrate it. Introducing the concepts of maximum and minimum allows us to introduce some important theorems, notably the intermediate value theorem and the fixed point theorem. These theorems are essential in the study of functions. Finally, we come to the definition of derivability and differentiability. We give some interpretations of these two definitions, and demonstrate their equivalence. These discussions result in the construction of the derivative function. We study this function in detail, in particular the algebraic operations on these functions. We continue our study of the derivability of functions. We present the properties of derivable functions: the composition derivative of functions, Rolle's theorem and the finite increase theorem. We also look at whether the derived function is continuous, and give some examples and counter-examples. Finally, we show the interest of the theorem of finite increments, which is a generalization of Rolle's theorem. This theorem is very important, as it has implications for the monotonicity of a derivable function.