The Lebesgue-Cantor function
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Theorem
There exists a continuous
monotone function
which is constant on every interval in
Proof:
Let be a decreasing sequence of closed sets used in the construction
of the ternary Cantor set , so that
.
For , define
(the closure of the complement of ).
Note that
and so it consists of the closure of all
intervals removed from [0,1] up to the -th step in the construction of the Cantor
set.
It consists of disjoint intervals , ordered so that
.
Picture 1:
_______________
. ______ _______________ ______
. __ ______ __
_______________ __ ______ __ .
We define a sequence of monotone continuous
functions as
follows:
and
and
for
Picture 2:
Note that if , then
.
Thus the sequence of monotone continuous
functions ,
trivially converges on the set
and defines a continuous monotone function on this set. Since is dense in the function extends uniquely to the entire unit interval and has desired properties.