Re: constant and locally constant

On Tue, 26 Feb 2008, water wrote:

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

A calculus student might "solve" this by spotting that such a function
would have to have a zero derivative everywhere. Thus, by a familiar
calculus principle, the function is constant. But the proof of that
principle requires a compactness argument anyway, and the exercise
should be attempted from first principles.

It does? If f is locally constant at each point of (a,b), or of R, then
by the calculus argument, f is constant over (a,b), or R.

Above is copied from a paper.

I think "a familiar calculus principle" is that f is constant when f has
zero derivative everywhere.

Why does the principle need compactness?

It does? Here's from first principles independent of calculus, real
numbers and compactness. Using only that the domain is connected.

Let f:X -> Y be a locally constant function over a connected, space X.
For example R, (0,1), [0,1].

For all y, f^-1(y) is open. Proof. If x in f^-1(y), then
f(x) = y; some open U nhood x with f(U) = y
x in open U subset f^-1f(U) = f^-1(y)
As every point of f^-1 is in an open subset of f^-1(y), f^-1y) is open.

Now assume f(x) /= f(y).
Thus X = f^-1f(x) \/ { f^-1(a) | a /= f(x) }
is a disconnection of X by open sets. QED.

Give an example of a locally constant function that is not constant.
.

Relevant Pages

  • Re: constant and locally constant
    ... A calculus student might "solve" this by spotting that such a function ... But the proof of that principle requires a compactness ... If a differentiable function is not constant, ...
    (sci.math)
  • constant and locally constant
    ... A calculus student might "solve" this by spotting that such a function ... But the proof of that principle requires a compactness ... and the exercise should be attempted from first principles. ...
    (sci.math)
  • Calculus vs. Non-contradiction?
    ... principle in logic of Non-Contradiction to be correct. ... that the proposition of calculus disproves the Law of Non- ... Contradiction in logic. ...
    (sci.logic)