Suppose that f: R→ R is a function. Suppose that there exists a constantc< 1 so that for all x, y E R we have f(x) = f(y)| < cx - y. Prove thatf is continuous on R. Theorem 4.3.2 (Characterizations of Continuity). Let f: A → R, and letCE A. The function f is continuous at c if and only if any one of the followingthree 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) (andTEA) 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

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Suppose that f: R→ R is a function. Suppose that there exists a constant c< 1 so that for all x, y E R we have f(x) = f(y)| < cx - y. Prove that f is continuous on R.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.4: Related Rates

Problem 4E: Assume x and y are functions of t. Evaluate dydtfor each of the following. 4x36xy2+3y2=228;...

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Question

100%

Suppose that f: R→ R is a function. Suppose that there exists a constant
c< 1 so that for all x, y E R we have f(x) = f(y)| < cx - y. Prove that
f is continuous on R.

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

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