Re: constant and locally constant

On Feb 26, 9:12 pm, 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.

You are quite right, the previous poster omitted an important step:

(0) A locally constant function is continuous (using definitions).
.