# Countable discontinuity

Discussion in 'Undergraduate Math' started by SHEN, Jan 19, 2006.

1. ### SHENGuest

Hi,

I'm trying to prove that f(x), being bounded in the interval (a, b) and
sign-preserving (i.e. if f(x0)> or <0, then f is positive or negative in
some neighbourhood), has a countable set of discontinuities.

Could anyone give some hint?

SHEN.

SHEN, Jan 19, 2006

2. ### Torsten HennigGuest

Hi,
With the above assumptions on f, your statement is
wrong.
(Consider f(x) = 1, if x is rational, and f(x)=2 else)

Best wishes
Torsten.

Torsten Hennig, Jan 19, 2006

3. ### SHENGuest

You're right, but I'm still wondering the intension of the problem's author.

SHEN, Jan 19, 2006
4. ### David C. UllrichGuest

Well since the problem as you stated it is wrong, it's hard to see
what that means.

Possibly the problem as the author stated it is actually correct,
and you're misinterpreting what something means in the statement.
You could try posting a word-for-word statement of the problem,
_exactly_ as it appears in whatever source this came from.

(For example the statement would be true if f were monotone,
and so I wonder whether something that says that f is monotone
got mangled somehow into your "sign-preserving" condition.
One reason for wondering that is that I doubt that "sign-preserving"
is the terminology the author used for "if f(x0)> or <0, then f
is positive or negative in some neighbourhood", because
that terminology for that condition doesn't make much sense.
So I suspect you've rephrased some things...)

************************

David C. Ullrich

David C. Ullrich, Jan 20, 2006
5. ### SHENGuest

Sorry for late in response. Actually, it is the note of the author for
the term "sign-preserving". I decide to give up this wrong one and go on
the next.

SHEN, Jan 24, 2006