By Richard J. Bagby

Introductory research addresses the desires of scholars taking a path in research after finishing a semester or of calculus, and provides an alternative choice to texts that imagine that math majors are their purely viewers. by utilizing a conversational variety that doesn't compromise mathematical precision, the writer explains the fabric in phrases that support the reader achieve a less assailable seize of calculus thoughts. * Written in a fascinating, conversational tone and readable sort whereas softening the rigor and concept * Takes a practical method of the mandatory and available point of abstraction for the secondary schooling scholars * a radical focus of easy themes of calculus * encompasses a student-friendly advent to delta-epsilon arguments * contains a restricted use of summary generalizations for simple use * Covers traditional logarithms and exponential services * offers the computational suggestions frequently encountered in simple calculus

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We use induction to prove the existence of a sequence of intervals like we described above. Getting the first interval in the sequence is easy: [a1 , b1 ] = [a, b]. Given x ∈ [a, b] we can choose x1 = x ∈ [a1 , b1 ] to get f (x1 ) ≤ f (x), so [a1 , b1 ] has the property we’re looking for. To prove the existence of the en- 42 CHAPTER II CONTINUOUS FUNCTIONS tire sequence, we assume that [an , bn ] is a closed subinterval of [a, b] with length 21−n (b − a) and that for each x ∈ [a, b] there is an xn ∈ [an , bn ] with f (xn ) ≤ f (x).

Let g be the numerical function whose value at each x ∈ R is the decimal value of x, rounded to the nearest hundredth. Show that g has discontinuities by producing an a ∈ R and an ε > 0 for which no corresponding δ > 0 exists. How small does ε need to be in this case? 7. 005)? 3. THE INTERMEDIATE VALUE THEOREM 33 8. Suppose that f is a numerical function with domain E, and that a is a point of continuity of f . Prove that there is an open interval I containing a for which the set of values {f (x) : x ∈ I ∩ E} has a supremum and an infimum.

We call h the common extension of f and g to the union of their domains. What about the continuity of the common extension of two functions? There are some simple implications here that we’ll state in a formal theomax {a, b} = 38 CHAPTER II CONTINUOUS FUNCTIONS rem. Its proof is quite easy, chiefly because it fails to address the difficult cases. 1: For A and B subsets of R, suppose that we have two functions f : A → R and g : B → R whose values agree on A ∩ B . Then their common extension is continuous at each point where both f and g are continuous, and any discontinuity of their common extension must be a discontinuity for f or for g , if not for both.

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