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Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

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quasi

Posts: 10,226
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 5:08 AM
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William Elliot wrote:
>baclesback wrote:
>>
>> More specifically, use the representation of x in C Cantor
>> set in base 3 with only 0's and 2's in the expansion of 3,
>> and map
>>
>> f: x=0.a1a2..... ---> 0.b1b2.......
>>
>> Wheref(bi)= 0 , if ai=0 , f(bi)=1 , if ai=2 .
>>
>> This f is continuous, but not absolutely continuous ( which
>> preserves sets of measure zero. )

>
>Why is f continuous?


Because f is increasing.

If X,Y are subsets of R, and f: X -> Y is a monotonic function,
then f is continuous (with respect to the relative topologies on
X and Y inherited from R).

quasi



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