Intermediate Value Theorem

In this section we present a straightforward theorem, the intermediate-value theorem and show that it can be very powerful in the study of equilibrium, which is one of the most important concepts in economics. The particular application we make is a very simple one. The rtnge of applications, however, is in fact very broad.

Suppose that the function )' -- f(x) is continuous on the interval fa, ¿J. b > a. It follows that the function must take on every value between f(a) and /(/>). which are the function values at the endpoints of the interval fa. ¿J. This result is called the intermediate-value theorem because any intermediate value between f(a) and fib) must occur for this function for at least one value of x between x — a and x — b. This result is understood intuitively by looking at figures 4.23 and 4.24

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Figure 4.23 Demonstration of the intermediate-value theorem for a function f(x) that is continuous on |ü. b\

Figure 4.24 Demonstration that if fix) is discontinuous on \u. b\- then it doesn't necessarily lake on every value between /(«) and fib)

Figure 4.23 Demonstration of the intermediate-value theorem for a function f(x) that is continuous on |ü. b\

Figure 4.24 Demonstration that if fix) is discontinuous on \u. b\- then it doesn't necessarily lake on every value between /(«) and fib)

In figure 4.23 il is clear that any value between y = f{a) and y = fib), for example, y — y, is realized by the continuous function /(.r) for some x e [it. h]. In figure 4.24 we see that this is not necessarily the case for a discontinuous function as no value between y = Aandy = ti is realized by the function for any x e [«./>].

Theorem 4.2 iIntermediate-value theorem) Suppose that f{.c) is a continuous function on the closed interval [a, b\ and that f(d) /(£»). Then, for any number y between f(a) and fib). there is some value of \. say .v = c. between a and b such that y = f(0.

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