The first existence theorem

This section is devoted to the formulation of the main existence theorem of Part I. Its proof will be worked out in Sections 2–7 and summarized in Section 8. We shall lead up to its statement by examining a number of special cases. Recall that our problem is to formulate a criterion which will tell us in many cases whether or not an equation of the form f(x) = y can be solved for x. To see what form the criterion might take, we examine cases where we know how to solve the equation completely.

Consider first the function f(x) defined by the formula x2 + 1 for x-values between -1 and +2. (The formula makes sense for x-values outside the interval -1 to 2, but we shall ignore this fact.) The function can be pictured from its graph shown in Fig. 1.1. The equation y = x2 + 1 defines a parabola, and our graph is the piece of the parabolic curve between the vertical lines where x = -1 and x = 2.

Notice first that there is a lowest point on the curve at x = 0, y = 1. This can be restated precisely: x2 + 1 is greater than or equal to 1 for all x between -1 and 2, and it has the minimum value 1 when x = 0.