Consider a “smooth curve” on the plane such as a circle or a parabola. Intuitively, we think of such curves as smooth because they have a tangent line at each point and because the slopes of these tangent lines vary continuously as the point of tangency moves along the curve. The concept of the derivative enables us to discuss such geometric phenomena analytically. Thus, for example, we refer to the graph of a function f : (a, b) → ℝ as a smooth curve when f has a continuous derivative on (a, b). The reader will be familiar with such applications of the notion of the derivative. In particular, we assume that the reader knows how to calculate the derivatives of trigonometric functions as well as how to use derivatives to sketch graphs and compute maxima and minima. More sophisticated applications of derivatives involve complex interactions of differentiation (the operation of taking the derivative of a function) with other mathematical operations (e.g., infinite summation and integration). For this, it is essential that we have a precise understanding of the definition of the derivative and that we develop the skills necessary for investigating the theoretical properties of differentiable functions. This is the goal of the present chapter.

Baire category theorem

In the space 〈ℝ, O〉 some sets may be regarded as “thin” and some as “thick.” For example, finite sets and the set ℤ are thin but intervals and open sets are not. The Baire category theory provides a precise definition of these concepts and includes the result that, in a complete metric space, each nonempty open set is a “thick” set.

The historical roots of Baire category theory lie in the last two decades of the nineteenth century and are associated with the characterization of the set Df of discontinuity points of a Riemann integrable function f. The question was, how thick could Df be. It is not perhaps an exaggeration to assert that modern mathematical analysis evolved from two strands of investigations into that question. In one strand, the notion of a measure-zero set (see Definition 7.4.8) was discovered; the other strand started from the notion of a nowhere-dense set (defined below) and sought to find a topological characterization of Df. The first strand led to the creation of modern integration theories and the second gave rise to the subject of point-set topology, which provides a framework for mathematical theories that treat functions as points in an abstract space with geometric and algebraic properties.

The discrete and the continuous are among the most fundamental categories of the human mind, and our urge to create theories that connect the two has prompted us to invent and deploy the infinite set in a monumental intellectual endeavor known as mathematical analysis.

As a branch of mathematics, analysis has evolved during the last four centuries. Prior to this time, mathematics was mainly geometry and arithmetic (together with some algebra). Natural numbers were the primary concepts of arithmetic, which provided for a quantitative study of discrete phenomena; and straight lines, curves, surfaces, etc. were the primary concepts of geometry, which provided for a quantitative study of continuous phenomena. So from a historical point of view, an understanding of the continuous in terms of the discrete could mean none other than constructing analytic models of the primary concepts of geometry using the stuff of arithmetic, which we accomplished under the auspices of our infinite sets.

The familiar real number system is one example of an analytic model of the geometric line, and the familiar system of the complex numbers is one example of an analytic model of the plane.