UCLA MATH 131A - quiz3 (3 pages)

Math 131A Real Analysis Quiz 3 Instructions You have 30 minutes to complete the quiz There are two problems worth a total of 20 points You may not use any books or notes Partial credit will be given for progress toward correct proofs Write your solutions in the space below the questions If you need more space use the back of the page Do not forget to write your name and UID in the space below Name Student ID number Question Points Score 1 10 2 10 Total 20 Problem 1 Suppose s 0 1 R t 1 2 R and u 2 4 R are continuous functions Suppose that s 1 t 1 and t 2 6 u 2 Let f 0 2 R be defined by s x f x t x if x 0 1 if x 1 2 Let g 1 4 R be defined by g x t x u x if x 1 2 if x 2 4 a 6pts Prove that f is continuous at 1 using the definition Solution Let 0 Since s is continuous at 1 there is a s 0 1 such that 1 s x 1 s x s 1 Since t is continuous at 1 there is a t 0 1 such that 1 x 1 t t x t 1 Let min s t Then 0 Suppose x 0 2 and x 1 Then either 1 x 1 in which case 1 s x 1 so that f x f 1 s x s 1 or 1 x 1 in which case 1 x 1 t so that f x f 1 t x t 1 b 4pts Prove that g is NOT continuous at 2 using the sequence definition 1 Solution Consider the sequence xn n 1 defined by xn 2 n We have limn xn 2 However using that xn 2 4 for each n N and the definition of g the fact that u is continuous at 2 and the definition of g again we see that lim g xn lim u xn u 2 6 t 2 g 2 n n Problem 2 a 5pts Suppose f R R is continuous at 0 and that f 0 6 0 Prove that x R f x 6 0 is an infinite set Help with a well chosen will help you ideas similar to 1 a and b of the homework helped me Solution Let f 0 Then 0 Since f is continuous at 0 we can find a 0 so that f x f 0 whenever x 0 i e f x f 0 f 0 whenever x This means x R f x 6 0 b 5pts Suppose that f 0 R is continuous at 0 and that f x 0 for all x 0 Prove that f 0 0 You may use any facts that were proved in the homework as long as you cite them when you use them Help One sequence really really helped me out 1 Solution Let xn n 1 be the sequence defined by xn n Since f is continuous at 0 each xn 0 and limn xn 0 we have f 0 lim f xn n Since xn 0 for all n N f xn 0 for all n N and so problem 1 b of the homework tells us that f 0 0