PlanetMath: differentiable function

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

differentiable function

(Definition)

Let $f\colon V\to W$ be a function, where and are Banach spaces. For $x\in V$ , the function is said to be differentiable at if its derivative exists at that point. Differentiability at $x\in V$ implies continuity at . If $S\subset V$ , then is said to be differentiable on if is differentiable at every point $x\in S$ .

For the most common example, a real function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable if its derivative $\frac{df}{dx}$ exists for every point in the region of interest. For another common case of a real function of variables $f(x_1,x_2,\ldots,x_n)$ (more formally $f\colon\mathbb{R}^n\to\mathbb{R}$ ), it is not sufficient that the partial derivatives $\frac{\partial f}{\partial x_i}$ exist for to be differentiable. The derivative of must exist in the original sense at every point in the region of interest, where $\mathbb{R}^n$ is treated as a Banach space under the usual Euclidean vector norm.

If the derivative of is continuous, then is said to be . If the th derivative of is continuous, then is said to be . By convention, if is only continuous but does not have a continuous derivative, then is said to be . Note the inclusion property $C^{k+1} \subset C^k$ . And if the -th derivative of is continuous for all , then is said to be $C^\infty$ . In other words $C^\infty$ is the intersection $C^\infty = \bigcap_{k=0}^\infty C^k$ .

Differentiable functions are often referred to as smooth. If is , then is said to be -smooth. Most often a function is called smooth (without qualifiers) if is $C^\infty$ or , depending on the context.

"differentiable function" is owned by Koro. [ full author list (3) | owner history (2) ]

(view preamble)

Other names:

smooth function, differentiable mapping, differentiable map, smooth mapping, smooth map, continuously differentiable

Also defines:

differentiable, smooth

Keywords:

differentiable, smooth

Attachments:: example of differentiable function which is not continuously differentiable (Example) by Koro
differentiable functions are continuous (Theorem) by matte

Log in to rate this entry.
(view current ratings)

Cross-references: intersection, property, inclusion, continuous, Euclidean vector norm, partial derivatives, sufficient, variables, region, real function, implies, point, derivative, Banach spaces, function
There are 333 references to this entry.

This is version 21 of differentiable function, born on 2002-05-19, modified 2006-06-08.
Object id is 2919, canonical name is DifferntiableFunction.
Accessed 40264 times total.

Classification:

AMS MSC:	57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings)
	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)

Pending Errata and Addenda

None.[ View all 9 ]

Discussion

forum policy

functions on a fin. dim. vector space by rmilson on 2002-06-09 00:36:03

In regards to the point raised in correction number 4.

Let's say V is an n-dimensional, real vector space.

1) Note that V is not a banach space, although it could be made
into one in an infinite number of ways.

2) It is true that V is isomorphic to R^n, in the sense
that there exist linear bijections between V and R^n,
however there is no way to prefer any one such
bijection over another.

We would like to define what it means for a
function f:V -> R to be differentiable, but how
to proceed?

We need a norm for the denominator of our limit
expression, but which norm are we to use?

The fact of the matter is that if f:V->R is differentiable
with respect to one norm, it is differentiable with
respect to all norms. This is an interesting consequence
of the finite-dimensionality of V.

Thus the concept of "differentiable function"
makes sense in the context of "finite dimensional
vector spaces", which is not quite the same context as
"Banach spaces".

A similar phenomenon occurs if we try to topologize
V. The way to proceed is to pick a norm, any norm
and to use that particular norm topology. It doesn't
matter which norm we choose to do this, we get the
same topology regardless. This is, of course,
not true in the infinite-dimensional
setting.

[ reply | up ]

Interact