By Michael Spivak

**Publish 12 months note:** First released in 1965

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This little ebook is mainly focused on these parts of ”advanced calculus” during which the subtlety of the innovations and strategies makes rigor tricky to achieve at an easy point. The technique taken the following makes use of hassle-free models of contemporary equipment present in refined arithmetic.

The formal necessities contain just a time period of linear algebra, a nodding acquaintance with the notation of set thought, and a good first-year calculus path (one which not less than mentions the least higher sure (sup) and maximum reduce certain (inf) of a collection of genuine numbers).

Beyond this a undeniable (perhaps latent) rapport with summary arithmetic might be came across nearly crucial.

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**Example text**

6. Clearly, we would not want to consider this to be continuous , so that we need a condition that describes this type of jump behaviour. 6: A jump discontinuity , while the values of for slightly less than 1 are close to 3 and the values of example, for slightly greater than 1 are close to 2. There would be no jump if values of were close to 2 (the value of ) for all close to 1. This gives us another criterion for continuity at a point. ❨ ❨ ✰ ✥ ✲ ✰ ✖ ✲ ✡ ❈ ❨ ✰ ✥ ✲ ✥ ✥ ❨ ❨ ✰ ✖ ✲ ✰ ✥ ✲ ✥ of its domain, it is necessary that Criterion 2 In order for a function to be continuous at a point the values of are close to the value of as long as is close to .

It should be clear how to deﬁne continuity on intervals such as or and we leave this as an exercise. A function which is continuous on an interval (closed, open or neither) is said to be continuous on . A function which is continuous at all points of its domain is simply referred to as being a continuous function . Thus polynomials are continuous functions. 2 In Exercises 1–6, the ﬁrst few terms of a sequence are given. Find an expression for the general term of each sequence, assuming the pattern continues as indicated.

1 be a sequence converging to Let ➔ ✥ (a) ↕ ➧ (b) ➔ ✥ ✥ ↕ ➔ ✥ ↕ ➔ ✥ ↕ ➔ ✥ ↕ ↕ ➔ (c) (d) (e) → converges to → ❞ ❇ ④ ⑦ ↕ ↕ ↕ → ④ ↕ → ✥ converges to ✥ ❺ and let ❺ for any constant ❺ converges to converges to → ✥ converges to ④ → ④ ➧ ✥ ④ ✥ ❺ ✥ ❺ ❞ ❇ ④ ❺ ④ ❺ ➔ ④ ↕ be a sequence converging to → ④ ❺ . Then ➧ ❺ ❺ ⑦ ④ ❺ if ④ ❺ ✡ ❦ and ✙ ④ ↕ ✡ ❦ for any . ✙ ❿ This theorem can be proved using a careful deﬁnition of convergence of sequences, but we shall not give the proof here. It is often possible to use this theorem to prove results on continuity, as we will show below.