Ask HN: A calculus theorem question?

by zeynel1 on 10/5/2010, 3:57 PM with 12 comments

Any curve can be considered to be made of sufficiently small straight lines. http://musr.physics.ubc.ca/~jess/hr/skept/Math/node10.html

What is the name of the theorem which states this fact? I know integral calculus sums small units of the curve; but what is the name of the theorem?

I asked the same in mathoverflow but it has been downvoted and got no answer. http://mathoverflow.net/users/9811/zeynel

Thank you

  • by drallison on 10/5/2010, 7:07 PM

    The problem is that your assertion is not true for any arbitrary function. For example, consider a function(aka curve) defined by: f(x) is x if x is irrational and 1 if x is rational. This function is discontinuous and cannot be approximated by shorter and shorter straight lines.

    The calculus concept you are reaching for is continuity. If f(x) is continuous, then for any e>0 there exists a d>0 such that abs( f(x) - f(y) ) < d if abs( x - y) < e. That's the famous epsilon-delta mantra of the calculus. See http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_....

    Continuity is a powerful property that enables much of what is useful in the calculus. Continuity of a function and its derivative is even stronger; continuity of the function and first and second derivatives is stronger still.

  • by rcfox on 10/5/2010, 4:13 PM

    It's not really a theorem; it's the definition of the derivative.

    f`(a) = lim h->0 (f(a+h) - f(a))/h

  • by eiji on 10/5/2010, 5:46 PM

    In mathematics you always try to minimize the preconditions, and the "theorem" you are looking for is likely a combination of strong special proofs.

    Maybe http://en.wikipedia.org/wiki/Trapezoidal_rule comes closest to what you are looking for. It states that you can approximate the definite integral using "straight lines", which includes your case, however, this rule has preconditions for the curve. HTH

  • by zeynel1 on 10/5/2010, 6:20 PM