Difference between revisions of "Differentiable"

Latest revision as of 14:32, 9 September 2007

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A real function $f$ is said to be differentiable at a point $P$ if $f$ is defined in an open neighborhood of $P$ and all partial derivatives of $f$ exist at $P$ . In particular, for a function $f$ defined on some subset $D$ of $\displaystyle \mathbb{R}$ taking values in $\mathbb{R}$ , $f$ is differentiable at $P \in D$ if and only if $D$ contains an open interval containing $P$ and the derivative of $f$ exists at $P$ .

A function $f: \mathbb R \to \mathbb R$ can fail to be differentiable at the point $\displaystyle x_0$ for the following reasons:

$f$ is not defined at $\displaystyle x_0$ , i.e. $\displaystyle f(x_0)$ doesn't exist.
$f$ is not defined on some set of points that includes members arbitrarily close to $\displaystyle x_0$ .
The derivative $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ is not defined at $\displaystyle x_0$ . Note that this requires at the very least that $\lim_{h \to 0}f(x_0 + h) = f(x_0)$ , i.e. any function differentiable at a point $\displaystyle x_0$ must also be continuous at that point.

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-A [[real number | real]] [[function]] $f$ is said to be '''differentiable''' at a point $P$ if $f$ is defined in an [[open set | open]] [[neighborhood]] of $P$ and all [[partial derivative]]s of $f$ exist at $P$.  In particular, for a function $f$ defined on some subset $D$ of $\mathbb R$ taking values in $\mathbb R$, $f$ is differentiable at $P \in D$ if and only if $D$ contains an open [[interval]] containing $P$ and the [[derivative]] of $f$ exists at $P$.
+A [[real number | real]] [[function]] <math>f</math> is said to be '''differentiable''' at a point <math>P</math> if <math>f</math> is defined in an [[open set | open]] [[neighborhood]] of <math>P</math> and all [[partial derivative]]s of <math>f</math> exist at <math>P</math>.  In particular, for a function <math>f</math> defined on some subset <math>D</math> of <math>\displaystyle \mathbb{R}</math> taking values in <math>\mathbb{R}</math>, <math>f</math> is differentiable at <math>P \in D</math> if and only if <math>D</math> contains an open [[interval]] containing <math>P</math> and the [[derivative]] of <math>f</math> exists at <math>P</math>.
-A function $f: \mathbb R \to R$ can fail to be differentiable at the point $x_0$ for the following reasons:
+A function <math>f: \mathbb R \to \mathbb R</math> can fail to be differentiable at the point <math>\displaystyle x_0</math> for the following reasons:
-* $f$ is not defined at $x_0$, i.e. $f(x_0) doesn't exist.
+* <math>f</math> is not defined at <math>\displaystyle x_0</math>, i.e. <math>\displaystyle f(x_0)</math> doesn't exist.
-* $f$ is not defined on some set of points that includes members [[arbitrarily close]] to $x_0$.
+* <math>f</math> is not defined on some set of points that includes members [[arbitrarily close]] to <math>\displaystyle x_0</math>.
-* The derivative $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ is not defined at $x_0$.  Note that this requires at the very least that $\lim_{h \to 0}f(x_0 + h) = f(x_0)$, i.e. any function differentiable at a point $x_0$ must also be continuous at that point.
+* The derivative <math>f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}</math> is not defined at <math>\displaystyle x_0</math>.  Note that this requires at the very least that <math>\lim_{h \to 0}f(x_0 + h) = f(x_0)</math>, i.e. any function differentiable at a point <math>\displaystyle x_0</math> must also be continuous at that point.
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Difference between revisions of "Differentiable"

Latest revision as of 14:32, 9 September 2007

See also