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Suppose that a < b are real numbers, and that f: [a, b] → R is a continu- ous function. Suppose also that for all x = [a, b] we have f(x) > 0. Prove f() is bounded. that the function } : [a, b] → R defined by () (x) = =

Suppose that a < b are real numbers, and that f: [a, b] → R is a continu- ous function. Suppose also that for all x = [a, b] we have f(x) > 0. Prove f() is bounded. that the function } : [a, b] → R defined by () (x) = =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 4.3.2 (Characterizations of Continuity). Let f: A → R, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e > 0, there exists a d>0 such that |x-c<8 (and x A) implies |f(x) = f(c) < €; (ii) For all Ve(f(c)), there exists a Vs (c) with the property that x E Vs(c) (and TEA) implies f(x) = V(f(c)); (iii) For all (n) →c (with En EA), it follows that f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). x-C
Transcribed Image Text:Theorem 4.3.2 (Characterizations of Continuity). Let f: A → R, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e > 0, there exists a d>0 such that |x-c<8 (and x A) implies |f(x) = f(c) < €; (ii) For all Ve(f(c)), there exists a Vs (c) with the property that x E Vs(c) (and TEA) implies f(x) = V(f(c)); (iii) For all (n) →c (with En EA), it follows that f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). x-C
Suppose that a < b are real numbers, and that f: [a, b] → R is a continu- ous function. Suppose also that for all x = [a, b] we have f(x) > 0. Prove f(T) is bounded. that the function } : [a, b] → R defined by () (x) = =
Transcribed Image Text:Suppose that a < b are real numbers, and that f: [a, b] → R is a continu- ous function. Suppose also that for all x = [a, b] we have f(x) > 0. Prove f(T) is bounded. that the function } : [a, b] → R defined by () (x) = =
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