Vipul: Created page with "{{function property}} ==Definition== ===Definition at a point=== Consider a function $f$ of one variable. We say that $f$ is '''differentiable''' at ..."

2011-10-16T16:16:43Z

Created page with "{{function property}} ==Definition== ===Definition at a point=== Consider a function <math>f</math> of one variable. We say that <math>f</math> is '''differentiable''' at ..."

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{{function property}}

==Definition==

===Definition at a point===

Consider a [[function]] <math>f</math> of one variable. We say that <math>f</math> is '''differentiable''' at a point <math>c \in \R</math> if the [[defining ingredient::derivative]] of <math>f</math> at <math>c</math> exists (as a finite number), i.e., we get a finite limit for the difference quotient.

Note that for a function to be differentiable ''at'' a point, the function must be defined on an open interval containing the point.

===Definition on an open interval===

Suppose <math>I</math> is an [[interval]] that is open but possibly infinite in one or both directions (i.e., an interval of the form <math>(a,b),(a,\infty),(-\infty,b),(-\infty,\infty)</math>). We say that <math>f</math> is differentiable on <math>I</math> if <math>f</math> is differentiable at every point of <math>I</math>.

===Definition on a union of open intervals===

A function is differentiable on a union of open intervals if it is differentiable on each of the open intervals.

===Definition without specification===

When we say that a function is differentiable (without specifying a point or interval or union of intervals), we mean that it is differentiable on its entire [[domain]] of definition. This makes sense if the domain is a union of open intervals.

For functions that are defined for all real numbers, this means that the function is ''everywhere'' differentiable, i.e., differentiable for all real numbers.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::continuously differentiable function]] || differentiable and the derivative is a continuous function || (by definition) || [[differentiable not implies continuously differentiable]] ||
|-
| [[Weaker than::multiply differentiable function]] (<matH>k</math> times differentiable for some positive integer <math>k</math>) || the function can be differentiated <math>k</math> times || || ||
|-
| [[Weaker than::infinitely differentiable function]] || || ||
|}
===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::continuous function]] || || [[differentiable implies continuous]] || [[continuous not implies differentiable]] ||
|}

Differentiable function - Revision history

Vipul: Created page with "{{function property}} ==Definition== ===Definition at a point=== Consider a function f of one variable. We say that f is '''differentiable''' at ..."

Vipul: Created page with "{{function property}} ==Definition== ===Definition at a point=== Consider a function $f$ of one variable. We say that $f$ is '''differentiable''' at ..."