HW9, Due November 7 1. Find the limits and justify your result using either the -δ or Sequential Criterion for limits 1 x→2 1 − x lim [[x]] = 2, lim x→e x2 x→0 |x| where [[x]] denotes the greatest integer less than or equal to x lim 2. Use the -δ definition of the limit to show that lim (x2 + 4x) = 12 x→2 and x+5 =4 x→−1 2x + 3 lim 3. Let c ∈ R and let f : R → R be such that limx→c (f (x))2 = L. (a) Show that if L = 0, then limx→c f (x) = 0 (b) Show by example, that if L 6= 0, then f may not have a limit at c. 4. Prove that the following functions defined on R are not continuous (a) The Dirichlet function from section 4.1 of Abott ( x sin( |x| ), x 6= 0 (b) The function f (x) = sin 1, otherwise What are the sets of discontinuities in the example above? (You may assume that the sine function is continuous) 5. Prove the following theorem Theorem 1. Let A ⊂ R and f : A → R. Let c ∈ R be the limit point of A. Then if limx→c f (x) > 0, there exists a neighborhood Vδ (c) of c, such that for all x ∈ A ∩ Vδ (c) and x 6= c, f (x) > 0. 6. Is the an analogue of Theorem 1 valid with limx→c f (x) ≥ 0? Prove or provide a counterexample. 7. Prove that the composition of continuous functions is continuous (Theorem 4.3.9 on p. 112 of Abott) √ √ 8. Prove that f (x) = n x is continuous. As with x the argument for x 6= 0 should be treated separately (as same ε-δ argument will not apply to both). The proof is broken into the following steps √ (a) Review the proof for the continuity of x. √ (b) Prove limx→0 n x = 0 P P1 n−j−1 y j (e.g. x2 − y 2 = (x − y) 1−j y j ) (c) Verify that xn − y n = (x − y) · n−1 j=0 x j=0 x √ √ √ (d) Prove that limx→c n x = n c (you may work out 3 x case separately as a warm up) 9. Given a function h : R → R define h−1 (A) = {x ∈ R : h(x) ∈ A}. Prove that the set h−1 (0) is closed for a continuous function h. 10. Generalize the previous example to a preimage of a closed set. That is if B ∈ R is closed, then h−1 (B) is closed for a continuous h : R → R. 1

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