w**a
发帖数: 1024 | 1 the set of continuous functions on an interval [0,1] is countable?
how to prove? | C********n
发帖数: 6682 | 2 dont think so
x^m m =[0,1]
【在 w**a 的大作中提到】
: the set of continuous functions on an interval [0,1] is countable?
: how to prove?
| l*****e
发帖数: 238 | 3 surely uncountable
define
f_a = x, for x \in [0,a]
a, for x > a
each of the above is continuous, and the collection {f_a} is uncountable,
due to the natural 1-1 correspondence with {a} for a \in [0,1]
【在 w**a 的大作中提到】
: the set of continuous functions on an interval [0,1] is countable?
: how to prove?
| j******w
发帖数: 690 | 4 has size 2^{\aleph_0}.
Just notice that every continuous function is decided by the
function restricted to rational numbers.
【在 w**a 的大作中提到】
: the set of continuous functions on an interval [0,1] is countable?
: how to prove?
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