Need somebody to work out these questions. deadline is tomorrow at noon

 

 

Math 320 Exam #2 due August 17, 2016 Name
Problem 1.(4 points) Let S ⇢ R be any set, and define for any x 2 R
the distance between x and the set S by
d(x, S) = inf{|x s| : x 2 S}. Prove the following:
(a) If x 2
/ S¯ then d(x, S) > 0.
(b) The function dS : R ! [0, +1), dS (x) = d(x, S), is Lipschitz continuous.
(c) If S is compact then for every x 2 R there is s 2 S such that
|x s| = d(x, S) Problem 2.(3 points) If f : E ! R is a continuous function defined on
some set E ⇢ R and if {aj }j2N ⇢ E is a Cauchy sequence is it true that
{f (aj )}j2N is also a Cauchy sequence ? If true prove it. Otherwise provide a
counterexample.
Problem 3.(4 points) Assume f : [0, 1] ! [0, 1] is a continuous function.
Show that there is x 2 [0, 1] with f (x) = x. Provide an example of a
continuous function g : (0, 1) ! (0, 1) where this is not true.
Problem 4.(4 points) Let f : (0, +1) ! R be a function which is twice
di↵erentiable and for which f 00 (x) c for all x > 0 and some constant c > 0.
Prove that f is not bounded from above.

Attachments:

• 1492672091-work.pdf