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(e) Show that va € R, Ve > 0,38 >0 such that f((a-8,a+5)) (f(a)- e. f(a)+c) is equivalent Ao the definition of f is continuous everywhere.

(e) Show that va € R, Ve > 0,38 >0 such that f((a-8,a+5)) (f(a)- e. f(a)+c) is equivalent Ao the definition of f is continuous everywhere.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.1: Inverse Functions
Problem 18E
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For this problem, let f: R→ R be a function. We say that a non-empty set UCR is ajar if Vu EU, 3r> 0 such that (u-r, u + r) CU. (a) Show that (0, 2) is ajar. (b) Find a set that is not ajar. You don't need to prove it. My set is In the following parts, we will use the follow two definitions. For A CR, we define f(A) = {f(a): a € A}. For BCR, we define f¹(B) := {r € R: f(r) € B} . Note that f(A) and f-¹(B) are both sets. (c) Show that va € R, Ve > 0,36 >0 such that f((a-d, a +5)) (f(a)- e, f(a) + e) is equivalent to the definition of f is continuous everywhere.
Transcribed Image Text:For this problem, let f: R→ R be a function. We say that a non-empty set UCR is ajar if Vu EU, 3r> 0 such that (u-r, u + r) CU. (a) Show that (0, 2) is ajar. (b) Find a set that is not ajar. You don't need to prove it. My set is In the following parts, we will use the follow two definitions. For A CR, we define f(A) = {f(a): a € A}. For BCR, we define f¹(B) := {r € R: f(r) € B} . Note that f(A) and f-¹(B) are both sets. (c) Show that va € R, Ve > 0,36 >0 such that f((a-d, a +5)) (f(a)- e, f(a) + e) is equivalent to the definition of f is continuous everywhere.
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