Consequences of Continuity and Differentiability

Transcription

1 Consequences of Continuity and Differentiability We have seen how continuity of functions is an important condition for evaluating limits. It is also an important conceptual tool for guaranteeing the existence of solutions of certain problems. The Intermediate Value Theorem If f (x) is a continuous function for x throughout the closed interval a x b, and y is any number between f (a) and f (b), then there is some c, a c b, for which f (c) = y. // That is, the graph of f must meet every horizontal line between y = f (a) and y = f (b). This helps with root finding: suppose that f is continuous and that f (a) and f (b) have opposite sign. The IVT implies that there must be a root of f in the interval a x b. In particular, if m is the midpoint of the interval (m = a +b ), then either m is 2 a root ( f (m) = 0) or f (m) has the same sign as either f (a) or f (b). But then f (m) has opposite sign from the other value. So the IVT applies again to force the existence of a root between m and that endpoint (either a or b). Repeating this process, we can successively bisect the interval between values of x at which f takes opposite sign at the endpoints to search for a root of f. Eventually, after applying

2 this bisection method often enough the distance between the endpoints of the interval are so small that we have a very good approximation of the position of the root. [Example 3, p. 278] [Example: p. 280, #39] A related fundamental existence theorem guarantees the existence of the solution to an optimization problem: The Extreme Value Theorem If f (x) is a continuous function for x throughout the closed interval a x b, then there must be some point in this interval where f takes a maxmimum value and some point in the interval where f takes a minimum value. // Note that the function need not be differentiable to have a maximum or minimum value; continuity is enough. But the condition that requires the interval to be closed is necessary. [See Example 4 (b,c), p. 279; also, #11, 13] The comment made in the last paragraph emphasizes the point that not all functions are either smooth or discontinuous; a function can be both continuous and nondifferentiable. Still, differentiability always implies continuity:

3 Theorem If f (x) is differentiable at x = a, then it is continuous there as well. Proof If f (x) is differentiable at x = a, then the limit f (a) = lim x a f (x) f (a) x a exists and is finite. But then, [ ] lim f (x) = lim f (x) f (a)+ f (a) x a x a f (x) f (a) = lim (x a)+ f (a) x a x a f (x) f (a) = lim lim (x a) x a x a x a = f (a) 0 + f (a) = f (a) [ ] + f (a) showing that f is continuous at x = a. //

4 A further important theoretical tool involving differentiability of a function comes in The Mean Value Theorem If f (x) is a continuous function for x throughout the closed interval a x b and is also differentiable throughout the open interval a < x < b, then there must be some point x = c in this interval where the derivative equals the slope of the line between the endpoints of the graph, that is, where f (c) = f (b) f (a). Proof Consider first the special case where f (a) = f (b). The theorem then claims that then there must be some point x = c where f (c) = 0; let s see why this should be true. Since the function is continuous, the EVT says that f has a maximum and minimum value somewhere in the interval a x b. If both of these points lie at the endpoints, then f (x min ) = f (a) = f (b) = f (x max ). But then the minimum and maximum are the same, so f is constant throughout the interval, and f (c) = 0 at every point x = c in the interval. Now if it is not the case that f (a) = f (b), let l(x) be the linear function that passes through the points (a, f (a)) and (b, f (b)). Then consider the new

5 function F(x) = f (x) l(x). Note that F(x) is continuous throughout the closed interval a x b and differentiable throughout the open interval a < x < b (as both f and l are), but also F(a) = f (a) l(a) = 0 = f (b) l(b) = F(b) so F is a function that satisfies the special case of the theorem that we have already proved. That is, then there must be some point x = c in the interval where F (c) = 0. But then 0 = f (c) l (c) = f (c) f (b) f (a) from which the conclusion of the theorem follows immediately. // [Example: p. 287, #19]

6 As a consequence of the MVT, we can assert the Theorem If f (x) = 0 for all x throughout an interval, then f (x) is a constant function. Proof Pick any two points in the interval, say x = a and x = b with a b. By the MVT, there must be some third point x = c with c between a and b so that f (c) = f (b) f (a). But because f (c) = 0, it follows that f (b) f (a) = 0, or simply that f (b) = f (a). As this is true for any two points in the interval, f is constant there. // Corollary Any two functions with the same derivative must differ by some additive constant. Proof Suppose F (x) = G (x) for all x (the underlying interval can be chosen arbitrarily). Then the function f (x) = F(x) G(x) satisfies the conditions of the last theorem, namely f (x) = 0 for all x throughout the interval. Consequently, f (x) = C for some constant, so F(x) = G(x)+C. // Another consequence is The Speed Limit Law If f (x) is a continuous function for x throughout the closed interval

7 a x b and is also differentiable throughout the open interval a < x < b, and if f (x) M for all x in the open interval, then f (b) f (a) M(). Proof If the conclusion were false, then we would have f (b) f (a) > M() instead, or f (b) f (a) > M. But by the MVT, there is some x = c in this interval f (b) f (a) where f (c) =. This means that f (c) > M. But this contradicts the hypothesis that f (x) M for all x in the interval. So the conclusion of the theorem must be true. // Recall that the Speed Limit Law is still valid (with the same proof) if we replace the appearances of the inequality in its statement with any of <, >, or.