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

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quasi

Posts: 10,403
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 5:31 AM
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quasi wrote:
>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.


I meant:

Because f is _strictly_ increasing.

>If X,Y are subsets of R, and f: X -> Y is a monotonic function,

I meant:

If X,Y are subsets of R, and f: X -> Y is a _strictly_ monotonic
function,

>then f is continuous (with respect to the relative topologies on
>X and Y inherited from R).


quasi



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