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