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4130 HOMEWORK 6 Due Thursday April 1 (1) Let A R. A point x A is called isolated if it is not a cluster point of A. (a) Can an open set have an

I need help with the last problem, number 6
4130 HOMEWORK 6 Due Thursday April 1 (1) Let A R . A point x A is called isolated if it is not a cluster point of A . (a) Can an open set have an isolated point? Can a closed set have one? (b) Give an example of a countable set with no isolated points. (2) Section 3.3.1 # 8. (3) Section 4.2.4 # 3. (Recall that an interval is, by definition, a subset I of R such that for all x,y I and all z R with x < z < y , we have z I .) (4) In this question, we will show that every positive real number has an n th root. (a) Let x (0 , ) and n N . Show that there exist α,β R with α n < x < β n . (b) Show that there exists y R with x = y n . (c) For x [0 , ), show that there exists a unique y [0 , ) with x = y 2 . We denote this y by x . (d) Define f : [0 , ) R by f ( x ) = x . Show that f is a continuous function. (5) Two monasteries, A and B , are joined by exactly one path AB which is 20 miles long. One morning, Brother Albert (a monk) sets out from monastery A at 9 am, arriving at monastery B at 9 pm. The next day, he sets out from monastery B at 9 am, arriving at monastery A at 9 pm. On both journeys, he may have stopped to rest, or even walked backwards for some of the time. (a) Prove that there is a point x on the path AB such that Brother Albert was at x at exactly the same time on both days. (Hint: let f i ( t ) be the distance of Brother Albert from A at time t on day i , i = 1 , 2. Apply the Intermediate Value Theorem to a suitable combination of f 1 and f 2 .) (b) Another monk, Brother Gilbert, has been dabbling in forbidden knowledge. Once per day, by snapping his fingers, he can instantaneously teleport him- self to any point within a 3 ft. radius of his current location. Suppose Brother Gilbert makes the same journey as Brother Albert. Does the conclusion from part (a) still hold? 1
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(6) Let f : R R be a function. Let a > 0. We say that f is periodic with period a if f ( x + a ) = f ( x ) for all x R . Suppose f : R R is periodic with period a and define g ( x ) = f (1 /x ) for x > 0. (a) Show that for all x > 0, we have f ([ x,x + a ]) = g ±• 1 x + a , 1 x ‚¶ . (b) Suppose f is not constant. Show that g is not uniformly continuous on (0 , ). [ Remark: In particular, taking f ( x ) = sin( x ) and a = 2 π , we see that sin(1 /x ) is not uniformly continuous.] 2
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