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Description
In this programme the concept of continuity is discussed and the theorem that a continuous function maps an interval to an interval is proved.
In this programme the concept of continuity is discussed and the theorem that a continuous function maps an interval to an interval is proved.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 10 |
| First transmission date: | 24-04-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:21:16 |
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| Contributors: | Colin Rourke; Peter Strain-Clark |
| Publisher: | BBC Open University |
| Keyword(s): | Closed Intervals; Continuous; Functions; Image Sets; Mapping |
| Footage description: | Colin Rourke introduces the programme by discussing the idea of continuity. He defines continuity in terms of nests of intervals converging on a spot in the domain giving rise to ja sequence of image sets which also form a nest. He then shows how this definition overcomes problems presented by functions with breaks in them. Peter Strain now gives a precise proof of the property that a function that's continuous map intervals to intervals. He does this by constructing a bisection nest, using the completeness axiom and using the definition of continuity. Colin Rourke discusses the need to extend these ideas to functions from R² to R. This involves defining a nest in R². Peter Strain uses this definition to create a test nest in R for the function mapping the point (x,y) to the point (x + y), and then finds the least closed images of these rectangles in the codomain. He then shows that as these intervals also form a nest then the function is continuous at every joint in the domain R . He uses the same technique to show that the product function is continuous. Colin Rourke now applies this idea to functions from R to R² showing that the function is contnuous if the two co-ordinate functions are continuous on their own. He then sums up the findings of the programme. |
| Master spool number: | HOU3252 |
| Production number: | FOUM055A |
| Videofinder number: | 4097 |
| Available to public: | no |
