Assume that ƒ (X) is a continuous real valued function on [0,1] such that f (0) = 6 and f(1) = 0. Prove the following: a. f(X) is uniformly continuous function on [0, 1] . b. Show that f([0,1]) closed and bounded. c. f([0,1]) is connected set. d. Show that there exists x, E (0,1) such that f(xo ) = 5.

Assume that ƒ (X) is a continuous real valued function on [0,1] such that f (0) = 6 and f(1) = 0. Prove the following: a. f(X) is uniformly continuous function on [0, 1] . b. Show that f([0,1]) closed and bounded. c. f([0,1]) is connected set. d. Show that there exists x, E (0,1) such that f(xo ) = 5.

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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Question

Assume that f(X) is a continuous real valued function
on [0,1]
such that f (0) = 6 and f(1) = 0.
Prove the following:
a. f(X) is uniformly continuous function on [0,1] .
b. Show that f([0,1]) closed and bounded.
c. f([0,1]) is connected set.
d. Show that there exists x, E (0, 1) such that
f(xo ) = 5.

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