The set of points which do not belong to the interior or exterior of A.

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What is an exterior point, ext(A)?

The interior complement of A, ie. int(A‹).

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What can the closure of A be written as?

The union of the interior and boundary of A, ie. Ā = A° U b(A).

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What is a convergent sequence?

The sequence <a1, a2, …> of real numbers converges to b e R; if for every € > 0, there exists a positive integer n• such that n > n• implies )an - b( < €.

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What is the intermediate value theorem?

If a function is continuous on a closed interval and u is a value between f(a) and f(b), then there exists c e [a,b] such that f(c) = u.

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What is the extreme value theorem?

In every interval [a,b] where a function is continuous, there is at least one maximum and one minimum; ie. it has at least two extreme values.

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Prove that f is continuous iff. for every open set V, f^-1 is open.

Suppose f is continuous on A. f^-1 is open in A. But A is open in R, and f^-1(V) = f^-1(V) ∏ A. So f^-1 is open.

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Proving backwards...

If f^-1(V) is open in R, then f^-1 ∏ A is open in A. f is defined from A to R, so preimage of any set is contained in A, so f^-1(V) ∏ A = f^-1(V). So f^-1(V) is open.

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