Metric and Topological Spaces

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## Revision of real analysis

The basic notions of analysis for R (= a complete ordered field) are :

1. A sequence (an) in R is convergent to  R if:
Given > 0 N N such that n > N |an - | < .

Informally: thinking of the terms of the sequence as approximations to the limit, the approximation gets better as you go further down the sequence.
For such a sequence we write (an)  .

2. A function is continuous at p R if:
Given > 0  > 0 such that |p - x| <  |f(p) _ f(x)| < .

Informally, points close enough to p are mapped close to f(p). By a continuous function we mean one which is continuous at all points where it is defined.

If you can draw the graph of a function, you should be able to spot whether it is continuous it will not, but functions defined in complicated ways this may be very hard to decide about.

In the next section we look at the first important generalisation.

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JOC February 2004