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'differentiable function'
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| Title of object: |
differentiable function |
| Canonical Name: |
DifferntiableFunction |
| Type: |
Definition |
| Created on: |
2002-05-19 03:20:51 |
| Modified on: |
2003-12-21 13:43:05 |
| Classification: |
msc:57R35, msc:26A24 |
| Keywords: |
differentiable, smooth |
| Defines: |
differentiable, smooth |
| Synonyms: |
differentiable function=smooth function
differentiable function=differentiable mapping
differentiable function=differentiable map
differentiable function=smooth mapping
differentiable function=smooth map
differentiable function=continuously differentiable |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\def\x{{\mathbf x}}
\def\sse{\subseteq}
\def\R{{\mathbb R}}
\def\del{\partial} |
Content:
Let $f\colon V\to W$ be a function, where $V$ and $W$ are Banach spaces%
\footnote{A Banach space has just enough structure for differentiability
to make sense. See the \PMlinkname{derivative}{Derivative2} entry for details.} %
(the most familiar case is when $f$ is a real function, that is $V=W=\R$;
$V$ and $W$ could also be differentiable manifolds). For $x\in V$,
the function $f$ is said to be {\em differentiable} at $x$ if its derivative exists at
that point. Differentiability at $x\in V$ implies continuity at $x$.
If $S\sse V$, then $f$ is said to be differentiable on $S$ if $f$ is
differentiable at every point $x\in S$.
For the most common example, a real function $f\colon\R\to\R$ 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 $n$ variables
$f(x_1,x_2,\ldots,x_n)$ (more formally $f\colon\R^n\to\R$),
it is not sufficient that the partial derivatives
$\frac{\del f}{dx_i}$ exist for $f$ to be differentiable. The
derivative of $f$ must exist in the original sense
at every point in the region of interest,
where $\R^n$ is treated as a Banach space under the usual Euclidean vector
norm.
If the derivative of $f$ is continuous, then $f$ is said to be $C^1$. If
the $k$th derivative of $f$ is continuous, then $f$ is said to be $C^k$. By convention, if $f$
is only continuous but does not have a continuous derivative, then $f$ is said to
be $C^0$. Note the inclusion property $C^k+1 \sse C^k$.
And if the $k$the derivative of $f$ is continuous for all $k$,
then $f$ 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 {\em smooth}. If $f$ is
$C^k$, then $f$ is said to be $k$-smooth. Most often a function is called
smooth (without qualifiers) if $f$ is $C^\infty$ or $C^1$, depending on the
context. |
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