Prove that there is a countable subset Eo of E such that f is 1-1 from E \ Eo onto [0,1]; i.e., prove that E is uncountable.

(d) Extend f from E to [0,1] by making f constant on the middle thirds Ek - 1 \

Ek . Prove that f : [0,1] --4 [0,1] is continuous and increasing. (Note: The function f is almost everywhere constant on [0,1], i.e., constant off a set of measure zero. Yet, it begins at f(O) = ° and ends at f(l) = 1.)