notes6 - CM221A ANALYSIS I NOTES ON WEEK 6 There will be no...

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CM221A ANALYSIS I NOTES ON WEEK 6 There will be no lectures on the week starting 8 November. Use the opportunity to revise the material covered in lectures. Many of you need to go through solutions to the class test and exercise sheets to see what you got wrong and why. DO NOT LEAVE THIS UNTIL THE SPRING VACATION! CONTINUOUS FUNCTION II A function f defined on an interval I is continuous if lim x c f ( x ) = f ( c ) for all c I . If c is an end point of I and c is included in I then we consider the right or the left limit at c . If c does not belong to I , we do not impose any condition on the right or left limit of f ( x ) at c . Clearly, if f is continuous on I then it is also continuous on any smaller interval I 1 I . Definition. We say that a function is bounded if its range (the set of its values) is a bounded subset of R . If a function f is bounded then its range has finite g.l.b. m and l.u.b. M and m 6 f ( x ) 6 M for all x from the domain of definition. The numbers m and M are called the greatest lower bound and, respectively,the least upper bound of the function f . Note that the function f may not attain the maximum value M and/or the minimum value m , even if it is continuous. Example. The function f ( x ) = (1 - x ) sin(1 /x ) is continuous and bounded on (0 , 1] and it has the least upper bound 1, but it does not take the value 1 anywhere, and there is no obvious way of defining it at x = 0. Example.
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notes6 - CM221A ANALYSIS I NOTES ON WEEK 6 There will be no...

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