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PROBLEM SET IV “BY EDWARD NASHTON” DUE FRIDAY, 7 OCTOBER Prove or give counterexamples to justify your claims. Exercise 29. Is there a continuous function f : R closed interval? Is there a continuous function f : R open interval? R such that the image of some closed interval is not also a R such that the image of some open interval is not also an R by the formula j1 k u(s) := 2 s − +s . 2 Is u continuous? Consider the function v : R R defined by the formula ( u(1/t ) if t 6= 0; v(t ) := 0 if t = 0, Exercise 30. Define the function u : R and the function w : R R defined by the formula ( w(t ) := t u(1/t ) if t = 6 0; 0 if t = 0, Is v continuous? Is w? Definition. A subset E ⊂ R is said to be closed if its complement R − E is open. R a continuous function. Must there be a function Exercise 31. Suppose E ⊂ R a closed set, and suppose f : E F:R R such that for any x ∈ E one has F (x) = f (x)? What if E were only assumed open? Exercise 32. Suppose E ⊂ R. Is the function dE : R R defined by the formula dE (x) := inf{|x − y| | y ∈ E} continuous? Exercise 33. Suppose E, E 0 ⊂ R two disjoint closed subsets. Must there be a continuous function f : R that both f −1 (0) = E and f −1 (1) = E 0 ? Exercise? 34. Let a < b be two real numbers. Is there a discontinuous function f : (a, b ) the sense that for any x, y ∈ (a, b ) and any t ∈ [0, 1], one has R such R that is convex, in f ((1 − t )x + t y) ≤ (1 − t ) f (x) + t f (y)? Exercise 35. Are there positive numbers r, s , t ∈ R with r ≥ t such that the limit x s + t 1/x lim x→0 r does not exist? Are there positive numbers r, s, t ∈ R with r ≥ t such that this limit exists and is greater than 1? 1