(iii) f is sequentially continuous at 0.
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.CR: Chapter 6 Review
Problem 2CR
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Please only answer part iii. Please can i have a written working out for this question. Thank You!
![(a) Define a function f: R → R by
f(x) =
=
{²
if x = Z
if x ¤ R \ Z.
Prove directly from the definition that:
(i) f is sequentially continuous at all a € R \ Z.
(ii) ƒ is sequentially discontinuous at all a € Z \ {0}.
(iii) f is sequentially continuous at 0.
(b) Prove directly from the definition that f(x) = x² is continuous.
(c) Prove that the equation
x + √√√x = 1
has a solution for x > 0.
(d) Show that the conclusion of the Intermediate Value Theorem can fail if the
function is not assumed to be continuous.
In other words, find a function f: [a, b] → R with f(a) < f(b), and a
vɛ (ƒ(a), f(b)) such that there does not exist any c = [a, b] with f(c) = v.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c5136e6-35b5-420e-8d27-15931b5cbc7e%2F01592b0c-d4ad-48a9-9bdf-091fa9b86d83%2F74i1e4o_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Define a function f: R → R by
f(x) =
=
{²
if x = Z
if x ¤ R \ Z.
Prove directly from the definition that:
(i) f is sequentially continuous at all a € R \ Z.
(ii) ƒ is sequentially discontinuous at all a € Z \ {0}.
(iii) f is sequentially continuous at 0.
(b) Prove directly from the definition that f(x) = x² is continuous.
(c) Prove that the equation
x + √√√x = 1
has a solution for x > 0.
(d) Show that the conclusion of the Intermediate Value Theorem can fail if the
function is not assumed to be continuous.
In other words, find a function f: [a, b] → R with f(a) < f(b), and a
vɛ (ƒ(a), f(b)) such that there does not exist any c = [a, b] with f(c) = v.
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