Re: constant and locally constant

On Tue, 26 Feb 2008, [ISO-8859-1] Mariano Suárez-Alvarez wrote:
On Feb 27, 1:12 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Tue, 26 Feb 2008, G. A. Edgar wrote:
water <waterloo2...@xxxxxxxxx> wrote:

Exercise 1. Show that a function that is locally constant at each
point of an interval [a,b] is constant on [a,b].

Why does the principle need compactness?

It doesn't "need" compactness, as the other replies show.
However, the simplest known proof does use compactness...
(1) A continuous function on a closed interval achieves a max and min
there (using compactness)

The problem does not grant that the funciton is continuous.

A function is continuous iff it is locally continuous,
essentially by definition. On the other hand, a locally
constant function is locally continuous, again essentially
by definition.

f:X -> Y is locally constant when for all x,
some c in Y, open U nhood x with f(U) = {c}.

f:X -> Y is locally continuous when for all x,
some open U nhood x with f|U in C(U,Y).

Ok, by the pasting lemma, locally continuous implies continuous.

Clearly on the other hand, locally constant implies locally continuous.

.

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