Difference between revisions of "Differentiable function"
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approaches a [[limit]]ing value that we call the [[derivative]] of the function f(x) at ''x=a''. | approaches a [[limit]]ing value that we call the [[derivative]] of the function f(x) at ''x=a''. | ||
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| + | There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point ''a'' if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if | ||
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| + | ::<math> |x - a| < \delta\,</math> | ||
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| + | then | ||
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| + | ::<math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>. | ||
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[[category:calculus]] | [[category:calculus]] | ||
Revision as of 04:55, June 19, 2011
A function f(x) is differentiable at the point a if and only if, as x approaches a (which it is never allowed to reach), the value of the quotient:
- <math>\frac{f(x) - f(a)}{(x - a)}</math>
approaches a limiting value that we call the derivative of the function f(x) at x=a.
There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point a if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if
- <math> |x - a| < \delta\,</math>
then
- <math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>.