# Countable Sets

Discussion in 'Undergraduate Math' started by SHEN, Jan 24, 2006.

1. ### SHENGuest

Hi, everyone,

Another difficulty to prove the following set to be countable,

1. f(x) is a real function defined on R^1.
2. for all x0 in R^1, exists delta>0, f(x)>=f(x0) holds when |x-x0|<delta.
Prove: f(R^1) is countable.

SHEN

SHEN, Jan 24, 2006

2. ### Brian M. ScottGuest

On Tue, 24 Jan 2006 22:31:56 +0800, SHEN
Let S = f(R^1), and suppose that S is uncountable. For each
y in S choose x(y) in R^1 such that y = f(x(y)), and let A =
{x(y) : y in S}. Show that for each x in A there are
rational numbers p(x) and q(x) such that p(x) < x < q(x),
and f(u) >= f(x) whenever p(x) < u < q(x). Conclude that
there are distinct x and x' in A such that p(x) = p(x') and
q(x) = q(x'). Finally, consider f(x) and f(x') to get a

Brian

Brian M. Scott, Jan 24, 2006

3. ### Brian M. ScottGuest

On Tue, 24 Jan 2006 17:20:52 -0500, "Brian M. Scott"
By the way, the converse is also true, and a nice little
problem. Let S be any countable set of real numbers. Then
there is a function f satisfying your conditions (1) and (2)
and having the further property that f(R^1) = S.

Brian

Brian M. Scott, Jan 25, 2006