Vipul: /* Statement for multiple differentiability */

2011-10-16T16:47:33Z

Statement for multiple differentiability

← Older revision		Revision as of 16:47, 16 October 2011
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	===Differentiable a fixed finite number of times===		===Differentiable a fixed finite number of times===

	All the above versions hold if we replace ''differentiable'' functions by <math>k</math> times differentiable functions, where <math>k</math> is a fixed positive integer. The interval version reads as follows:		All the above versions hold if we replace ''differentiable'' functions by [[fact about::multiply differentiable function]]s, i.e., <math>k</math> times differentiable functions, where <math>k</math> is a fixed positive integer (this means that the <math>k^{th}</math> derivative -- see [[higher derivative]] -- exists). The interval version reads as follows:

	For any interval <math>I</math> and positive integer <math>k</math>, the <math>k</math> times differentiable functions on <math>I</math> form a [[real vector space]], in the following sense:		For any interval <math>I</math> and positive integer <math>k</math>, the <math>k</math> times differentiable functions on <math>I</math> form a [[real vector space]], in the following sense:

Vipul: Created page with "==Statement for single differentiability== ===Differentiability at a point version=== ===Differentiability around a point version=== ===Differentiability on an open interval v..."

2011-10-16T16:08:25Z

Created page with "==Statement for single differentiability== ===Differentiability at a point version=== ===Differentiability around a point version=== ===Differentiability on an open interval v..."

New page

==Statement for single differentiability==

===Differentiability at a point version===

===Differentiability around a point version===

===Differentiability on an open interval version===

Consider an [[interval]] <math>I</math> that is open but possibly infinite in one or both directions (i.e., an interval of the form <math>(a,b),(-\infty,b),(a,\infty),(-\infty,\infty)</math>). A [[differentiable function]] on <math>I</math> is a function whose [[derivative]] exists at every point of <math>I</math>.

The differentiable functions on <math>I</math> form a [[real vector space]], in the following sense:

* '''Additive closure''': If <math>f</math> and <math>g</math> are differentiable functions on <math>I</math>, the [[pointwise sum of functions]] <math>f + g</math> is also differentiable on <math>I</math>.
* '''Scalar multiples''': If <math>f</math> is a differentiable function on <math>I</math> and <math>\lambda \in \R</math>, then <math>\lambda f</math> is a differentiable function on <math>I</math>.

==Statement for multiple differentiability==

===Differentiable a fixed finite number of times===

All the above versions hold if we replace ''differentiable'' functions by <math>k</math> times differentiable functions, where <math>k</math> is a fixed positive integer. The interval version reads as follows:

For any interval <math>I</math> and positive integer <math>k</math>, the <math>k</math> times differentiable functions on <math>I</math> form a [[real vector space]], in the following sense:

* '''Additive closure''': If <math>f</math> and <math>g</math> are <math>k</math> times differentiable functions on <math>I</math>, the [[pointwise sum of functions]] <math>f + g</math> is also <math>k</math> times differentiable on <math>I</math>.
* '''Scalar multiples''': If <math>f</math> is a <math>k</math> times differentiable function on <math>I</math> and <math>\lambda \in \R</math>, then <math>\lambda f</math> is a <math>k</math> times differentiable function on <math>I</math>.

===Infinitely differentiable===

The above also holds if we replace differentiable by the notion of [[infinitely differentiable function|infinitely differentiable]]. An infinitely differentiable function is a function that is <math>k</math> times differentiable for all <math>k</math>. The interval version reads as follows:

For any interval <math>I</math>, the infinitely differentiable functions on <math>I</math> form a [[real vector space]], in the following sense:

* '''Additive closure''': If <math>f</math> and <math>g</math> are infinitely differentiable functions on <math>I</math>, the [[pointwise sum of functions]] <math>f + g</math> is also infinitely differentiable on <math>I</math>.
* '''Scalar multiples''': If <math>f</math> is infinitely differentiable function on <math>I</math> and <math>\lambda \in \R</math>, then <math>\lambda f</math> is infinitely differentiable function on <math>I</math>.

Differentiable functions form a vector space - Revision history

Vipul: /* Statement for multiple differentiability */

Vipul: Created page with "==Statement for single differentiability== ===Differentiability at a point version=== ===Differentiability around a point version=== ===Differentiability on an open interval v..."