AoPSWiki
Visit the AoPS Book Store.
Personal tools

Differentiable

From AoPSWiki

This article is a stub. Help us out by expanding it.

A real function f is said to be differentiable at a point P if f is defined in an open neighborhood of P and all partial derivatives of f exist at P. In particular, for a function f defined on some subset D of \displaystyle \mathbb{R} taking values in \mathbb{R}, f is differentiable at P \in D if and only if D contains an open interval containing P and the derivative of f exists at P.

A function f: \mathbb R \to \mathbb R can fail to be differentiable at the point \displaystyle x_0 for the following reasons:

  • f is not defined at \displaystyle x_0, i.e. \displaystyle f(x_0) doesn't exist.
  • f is not defined on some set of points that includes members arbitrarily close to \displaystyle x_0.
  • The derivative f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} is not defined at \displaystyle x_0. Note that this requires at the very least that \lim_{h \to 0}f(x_0 + h) = f(x_0), i.e. any function differentiable at a point \displaystyle x_0 must also be continuous at that point.

See also

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Differentiable"
Want to learn how to tackle those tough MATHCOUNTS and AMC counting and probability problems? Check out Art of Problem Solving's Introduction to Counting & Probability by David Patrick.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us