Homework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December

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1 Homework for MATH 4603 (Advanced Calculus I) Fall 2015 Homework 13: Due on Tuesday 15 December 49. Let D R, f : D R and S D. Let a S (acc S). Assume that f is differentiable at a. Let g := f S. Show that g is differentiable at a and that g (a) = f (a). 50. Find a function f : R R such that f is differentiable on R and such that f is not continuous at Let a, b R. Assume that a < b. Let I := [a, b]. Let f : I R be semi-increasing. Let c I. Assume that f is differentiable at c. Show that f (c) 0. Homework 12: Due on Tuesday 8 December 44. Let D R and let f : D R. Assume that #D 4. (That is, assume either that D is infinite or that D is finite with 4 or more elements.) Assume that f is strictly 4-monotone, i.e., assume, for all S D, that Show that f is strictly monotone. [ #S = 4 ] [ (f S) is strictly monotone ]. 45. Let p : R R be an even degree polynomial. Assume that lim [p(x)] =. Show x that lim [p(x)] =. x 46. Find an open cover of [1, 2) that has no finite subcover. 47. Let I := [1, 2). Find a sequence a I N such that, for any convergent subsequence b of a, we have: lim b / I. 48. Let D R and let a D. Show that acc (D\{a}) = acc D. No homework is due on Tuesday 1 December Homework 11: Due on Tuesday 24 November 41. Let f : R R be continuous and let U R be open. Show that f (U) is open. Recall that f (U) := {x R f(x) U}. 42. Let f : R R be continuous and let C R be closed. Show that f (C) is closed. Recall that f (C) := {x R f(x) C}. 43. Find a continuous f : R R and a sequentially compact K R such that f (K) is not sequentially compact. Recall that f (K) := {x R f(x) K}. Homework 10: Due on Tuesday 17 November

2 39. Let A, B R. Assume that A and B are both connected. Show that A B is connected. 40. Find A, B R such that A and B are both connected and such that A\B is not connected. Homework 9: Due on Tuesday 10 November 36. Let D R, f : D R and a D. Assume that f is discontinuous at a. Show that f is discontinuous at a. 37. Let E R, D E, g : E R and a D. Assume that g is continuous at a. Show that g D is continuous at a. 38. Let I be an interval. Let f : I R be a semi-increasing function. Show that there exist an open interval J R and a function g : J R such that (i) I J, (ii) g I = f and (iii) g is semi-increasing. 31. Find sets D, E R such that D E is not bounded above, D E is not bounded below, 0 (acc D) (acc E) and 0 / acc (D E). Homework 8: Due on Tuesday 3 November 32. Let f, g : R R. Assume that f at 2 +. Assume that g at 2 +. Show that fg at 2 +. Recall: ( f at 2 + ) means: ( both [ 2 acc ((dom [f]) (2, )) ] and [ N R, δ > 0 s.t., x dom [f], ( (2 < x < 2 + δ) (f(x) < N) ) ] ). In this problem, dom [f] = R, so 2 acc ((2, )) = acc ((dom [f]) (2, )). 33. Let f, g : R R. Assume that f at 3. Assume that g(( 4, 3)) is bounded above. Show that f + g at Let s R N. Let t be a tail of s. Let L R. Assume t L. Show s L. 35. Let D R, φ : D R, a acc D and L R. Let V be a neighborhood of a. Let D 0 := (D V )\{a}. Let φ 0 := φ D 0 : D 0 R. Assume that φ 0 L at a. Show that φ L at a. Recall that R := R {, }. Homework 7: Due on Tuesday 27 October For any x R, we define x = and x =.

3 For any x R, we define ( ) x = and x ( ) =. We define = and =. Then, for all x R, we have x = = ( ) x. For any a R N, by a we mean: For all M R, there exists j 0 N such that, for all j N, [ j j 0 ] [ a j > M ]. For any a R N, by a we mean: For all N R, there exists j 0 N such that, for all j N, [ j j 0 ] [ a j < N ]. For any a R N, for any x R, by a x we mean: For all ε > 0, there exists j 0 N such that, for all j N, [ j j 0 ] [ a j x < ε ]. 26. Let S, T R. Assume S T. Show that inf T inf S and that sup S sup T. 27. Let a R N. Show that liminf a limsup a. 28. Let a, b R N. Assume, for all j N, that a j b j. Show that liminf a liminf b and that limsup a limsup b. 29. Let a R N. Assume limsup a =. Show that there exists a subsequence b of a such that b. 30. Let S R, let x R and let ε > 0. Assume that S (x ε, x + ε) =. Show that (sup S) / (x ε, x + ε). Homework 6: Due on Tuesday 20 October 23. For any a R N, by a we mean: For all N R, there exists j 0 N such that, for all j N, [ j j 0 ] [ a j < N ]. Let a, b R N. Assume that a and that b. Show that (ab). 24. Let c R. Let c denote the constant sequence c, c, c,.... Show that c c. 25. Let b R N. Assume that b is bounded. Show that there exists M > 0 such that, for all j N, we have b j < M. Homework 5: Due on Tuesday 13 October 19. Let a, b R N. Assume that a is bounded. Assume that b. That is, assume, for all M R, that there exists j 0 N such that, for all j N, [ j j 0 ] [ b j > M ]. Assume, for all j N, that b j 0. Define c R N by c j = a j /b j. Show that c Let a, b R N and let x R. Assume both that a 1, a 2, a 3, a 4,... x and that b 1, b 2, b 3, b 4,... x.

4 Show that a 1, b 1, a 2, b 2, a 3, b 3, a 4, b 4,... x. 21. Find the set of accumulation points of {j + (1/k) j, k N\{1}}. 22. Describe a sequence a R N such that: for all x R, there exists a subsequence b of a such that b x. (Note: If your sequence a is correct, you get full credit, even if you don t provide a proof. Recall that b is a subsequence of a means: there exists a sequence m N N such that for all j N, m j < m j+1 and for all j N, b j = a mj ]. NOTE: This is the full homework assignment due Tuesday 13 October. No further problems will be assigned for that date. Homework 4: Due on Tuesday 6 October 16. Let a R N and assume that a is convergent. Show that a is bounded below. Hint: In class, we proved that a is bounded above. Modify that proof. 17. Define a R N by a j = (2j 1)/(j +1). Show, directly from the defintion, that a 2. (Do NOT use any properties of convergence; at this point, we haven t proved any.) 18. For any a R N, by a we mean: For all M R, there exists j 0 N such that, for all j N, [ j j 0 ] [ a j > M ]. Let s, t R N. Assume that s is bounded below and that t. Define u R N by u j = s j + t j. Show that u. Homework 3: Due on Tuesday 29 September 11. Let A and B be sets. Assume that there exists f A B such that f is an injective function from A to B. Prove that there exists g B A such that g is a surjective partial function from B onto A, i.e., such that g is a surjective function from a subset of B onto A. 12. Prove, by induction on n, that: for all n N, n j 2 = j=1 n(n + 1)(2n + 1) Prove that there exists a bijection from the interval (2, 3) onto the interval [4, 7). 14. Let A and B be sets. Let f : A B. Prove, for all S, T B, that f (S\T ) = [f (S)] \ [f (T )]. 15. Prove or disprove: For any sets A and B, for any function f : A B, for any S B, f (f (S)) = S.

5 Homework 2: Due on Tuesday 22 September 6. Let X := R, Y := R and g := {(b 3, b) b R}. Prove: For all x X, for all y, z Y, [ ((x, y) g) and ((x, z) g) ] [ y = z ]. 7. Compute [1/n, n]. [ ] 8. Compute R [1/n, n]. 9. Compute ( R \ [1/n, n] ). 10. Let R := {[1/n, (n 1)/n] n N, n 2}. Compute R, i.e., compute R. Homework 1: Due on Tuesday 15 September 1. Prove, for all sets X and Y, that X Y X. 2. Prove, for all sets S, T and U, that [(S U) and (T U)] [S T U]. R R 3. Prove, for all sets S, T and U, that [T U] [(S T ) (S U)]. 4. Prove, by truth table: [P and (Q or R)] [(P and Q) or (P and R)]. 5. Prove, for all sets A, B and C, that A (B C) = (A B) (A C).