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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 03:20:51-04 |
| Modified on: |
2002-05-28 10:40:29-04 |
| Classification: |
msc:57R35, msc:26A24 |
| Keywords: |
differentiable, smooth |
| Defines: |
differentiable, smooth |
| Synonyms: |
differentiable function=smooth function |
Preamble:
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\def\x{{\mathbf x}}
\def\sse{\subseteq}
\def\R{{\mathbb R}} |
Content:
Let $f:V\to W$ be a function, where $V$ and $W$ are Banach spaces (the most familiar case is when $f$ is a real function, that is $V=W=\R$). 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 $\x\in S$.
If the derivative of $f$ is continuous, then $f$ is said to be $C^1$. If $f$ has
$k$ continuous derivatives, 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 $C^0$. And if $f$ has infinitely many continuous derivatives, then $f$ is
said to be $C^\infty$.
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
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