differentiable function
Let be a function, where and are Banach spaces.
For , the function is said to be differentiable
at if its derivative
exists at that point. Differentiability at
implies continuity at . If , then is said to
be differentiable on if is differentiable at every point .
For the most common example, a real function is differentiable
if its derivative exists for every point in the region of
interest. For another common case of a real function of variables
(more formally ),
it is not sufficient that the partial derivatives
exist for to be differentiable. The
derivative of must exist in the original sense
at every point in the region of interest,
where 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 .
And if the -th derivative of is continuous for all ,
then is said to be . In other words is the
intersection
.
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 or , depending on the context.
Title | differentiable function |
Canonical name | DifferentiableFunction |
Date of creation | 2013-03-22 12:39:10 |
Last modified on | 2013-03-22 12:39:10 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 24 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 26A24 |
Classification | msc 57R35 |
Synonym | smooth function |
Synonym | differentiable mapping |
Synonym | differentiable map |
Synonym | smooth mapping |
Synonym | smooth map |
Synonym | continuously differentiable |
Related topic | OneSidedDerivatives |
Related topic | RoundFunction |
Related topic | ConverseTheorem |
Related topic | WeierstrassFunction |
Defines | differentiable |
Defines | smooth |