(Continuity). A function f AR is continuous at a point c A if, for all e > 0, there exists a d> 0 such that whenever |x- c < 8 (and x A) it follows that f(x)-f(c) < €. If f is continuous at every point in the domain A, then we say that f is continuous on A. Use the e-d definition of continuity to prove that the following functions are continuous at the indicated real number xo: (a) f: R→R, f(x) = 7x-2, at xo = 4; b) g: (0, ∞) → R, g(x) = 1, xo = 10; (c) h: R→R, h(x) = 3x² + 12, xo = 1.

If a function f(x) is continuous on the interval [a..b] and sign of f(a) = sign of f(b), then: There is a value c E [a.b] such that: f(c) = 0 1.e., there is a root c in the interval [a..b] true false

Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.

The graph of f(x) is given below. Select each true statement about the continuity of f (x) at x = -1. Select all that apply: -7-6-5-4-31 N 10 f(x) is continuous at x = -1. S A W HN -7 ņ T لنا + ú 60 f(x) is not continuous at x = -1 because it is not defined at x = -1. 1 2 3 4 5 6 7 f(x) is not continuous at x = -1 because lim f(x) does not exist. X--1 f(x) is not continuous at x = -1 because lim f(x) + f(-1).