constant and locally constant

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.

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?

Thanks in advance!
.

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)
  • Re: constant and locally constant
    ... A calculus student might "solve" this by spotting that such a function ... calculus principle, the function is constant. ... Why does the principle need compactness? ... Let f:X -> Y be a locally constant function over a connected, ...
    (sci.math)
  • Re: Help with exercise in Apostol
    ... But the choice of axioms here is ... Yes, true, I overstated my case somewhat -- the exercise is not all ... this kind relates better to a Modern Algebra course than to a Calculus ...
    (sci.math)
  • Re: continuous two-to-one function from R->R
    ... Ioannis wrote: ... >> This is an exercise in CALCULUS by Spivack, ...
    (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)
Quantcast