By Robert G. Bartle

This introductory textbook info the rigorous remedy of the elemental conception of features of 1 genuine variable. Topological strategies resembling open set and closed set were amassed jointly to supply a extra unified dialogue.

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A,b) Apply the Mean Value Theorem. (c) Apply Darboux’s Theorem to the results of (a) and (b). 3 The proofs of the various cases of L’Hospital’s Rules range from fairly trivial to rather complicated. 5, which deals with the case ∞/∞. This requires a more subtle analysis than the other cases. This section may be regarded as optional. Students are already familiar with the mechanics of L’Hospital’s Rules. Sample Assignment: Exercises 1, 2, 4, 6, 7(a,b), 8(a,b), 9(a,b), 13, 14. Partial Solutions: 1.

A) f (x) = 2 for x > 0; f (x) = 0 for −1 < x < 0; and f (x) = −2 for x < −1, (b) g (x) = 3 if x > 0; g (x) = 1 if x < 0; g (0) does not exist, (c) h (x) = 2|x| for all x ∈ R, (d) k (x) = (−1)n cos x for nπ < x < (n + 1)π, n ∈ Z; k (nπ) does not exist, (e) p (0) = 0; if x = 0, then p (x) does not exist. 9. If f is an even function, then f (−x) = lim [f (−x + h) − f (−x)]/h = − lim [f (x − h) − f (x)]/(−h) = −f (x). h→0 h→0 10. If x = 0, then g (x) = 2x sin(1/x√2 ) − (2/x) cos(1/x2 ). Moreover, g√(0) = lim h sin(1/h2 ) = 0.

9 implies that αc and ωc are in R[a, b]. Moreover, b a (ωc − αc ) = 2M (c − a) < ε when c − a < ε/2M . 3 b b c implies that f ∈ R[a, b]. Further, | a f − c f | = | a f | ≤ M (c − a). 12. Indeed, |g(x)| ≤ 1 for all x ∈ [0, 1]. Since g is continuous on every interval [c, 1] where 0 < c < 1, it belongs to R[c, 1] and the preceding exercise applies. 13. Let f (x) := 1/x for x ∈ (0, 1] and f (0) := 0. Then f ∈ R[c, 1] for every c ∈ (0, 1), but f ∈ / R[0, 1] since f is not bounded. 14. Use Mathematical Induction.

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