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Theorem 7. If f x is continuous in a; b and if f a A and f b B, then corresponding to any number C between A and B there exists at least one number c in a; b such that f c C. This is sometimes called the intermediate value theorem. Theorem 8. If f x is continuous in a; b and if f a and f b have opposite signs, there is at least one number c for which f c 0 where a < c < b. This is related to Theorem 7. Theorem 9. If f x is continuous in a closed interval, then f x has a maximum value M for at least one value of x in the interval and a minimum value m for at least one value of x in the interval. Furthermore, f x assumes all values between m and M for one or more values of x in the interval. Theorem 10. If f x is continuous in a closed interval and if M and m are respectively the least upper bound (l.u.b.) and greatest lower bound (g.l.b.) of f x , there exists at least one value of x in the interval for which f x M or f x m. This is related to Theorem 9.
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PIECEWISE CONTINUITY A function is called piecewise continuous in an interval a @ x @ b if the interval can be subdivided into a nite number of intervals in each of which the function is continuous and has nite right- and lefthand limits. Such a function has only a nite number of discontinuities. An example of a function which is piecewise continuous in a @ x @ b is shown graphically in Fig. 3-4 below. This function has discontinuities at x1 , x2 , x3 , and x4 .
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UNIFORM CONTINUITY Let f be continuous in an interval. Then by de nition at each point x0 of the interval and for any  > 0, we can nd  > 0 (which will in general depend on both  and the particular point x0 ) such that j f x f x0 j <  whenever jx x0 j < . If we can nd  for each  which holds for all points of the interval (i.e., if  depends only on  and not on x0 ), we say that f is uniformly continuous in the interval. Alternatively, f is uniformly continuous in an interval if for any  > 0 we can nd  > 0 such that j f x1 f x2 j <  whenever jx1 x2 j <  where x1 and x2 are any two points in the interval. Theorem.
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If f is continuous in a closed interval, it is uniformly continuous in the interval.
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FUNCTIONS, LIMITS, AND CONTINUITY
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Solved Problems
FUNCTIONS 3.1. Let f x x 2 8 x for 2 @ x @ 8. (a) Find f 6 and f 1 . (b) What is the domain of de nition of f x (c) Find f 1 2t and give the domain of de nition. (d) Find f f 3 , f f 5 . (e) Graph f x .
(a) f 6 6 2 8 6 4 2 8 f 1 is not de ned since f x is de ned only for 2 @ x @ 8. (b) The set of all x such that 2 @ x @ 8. (c) f 1 2t f 1 2t 2gf8 1 2t g 1 2t 7 2t where t is such that 2 @ 1 2t @ 8, i.e., 7=2 @ t @ 1=2. (d) f 3 3 2 8 3 5, f f 3 f 5 5 2 8 5 9. f 5 9 so that f f 5 f 9 is not de ned. (e) The following table shows f x for various values of x.
f (x)