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Let $f\colon \R\to\R$ be a function. We say that $f$ is of class $C^1$ if $f'$ exists and it's continuous.
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Let $f:\R\to\R$ be a function. We say that $f$ is of class $C^1$ if $f'$ exists and it's continuous.
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| We also say that $f$ is of class $C^n$ if its $n$-th derivative exists and is continuous (and therefore all other previous derivatives exist and are continuous too). |
We also say that $f$ is of class $C^n$ if its $n$-th derivative exists and is continuous (and therefore all other previous derivatives exist and are continuous too). |
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| The class of continuous functions is denoted by $C^0$. So we get the |
The class of continuous functions is denoted by $C^0$. So we get the foloowing relationship among these classes: |
| following relationship among these classes: |
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| \[ |
\[ |
| C^0\supset C^1\supset C^2\supset C^3 \supset \ldots |
C^0\supset C^1\supset C^2\supset C^3 \supset \ldots |
| \] |
\] |
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| Finally, the class of functions that have continuous derivatives of any order is denoted by $C^\infty$ and thus |
Finally, the class of functions that have continuous derivatives of any order is denoted by $C^\infty$ and thus |
| \[ |
\[ |
| C^\infty = \bigcap_{n=0}^\infty C^n. |
C^\infty = \bigcap_{n=0}^\infty C^n. |
| \] |
\] |
| It holds that any function that is differentiable is also continuous |
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| (see \PMlinkname{this entry}{DifferentiableFunctionsAreContinuous}). |
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| Therefore, $f\in C^\infty$ if and only if every derivative of $f$ exists. |
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| The previous concepts can be extended to functions $f\colon \R^m \to \R$, |
The previous concepts can be extended to functions $f:\R^m \to \R^n$, where $f$ being of class $C^n$ amounts to asking for all the partial derivatives of order $n$ are continuous. For instance, $f\colon\R^m\to \R^n$ being $C^2$ means that |
| where $f$ being of class $C^n$ amounts to asking for all the |
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| partial derivatives of order $n$ are continuous. |
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| For instance, $f\colon\R^m\to \R$ being $C^2$ means that |
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| \[ |
\[ |
| \frac{\partial^2 f}{\partial x_j\partial x_i} |
\frac{\partial^2 f}{\partial x_j\partial x_i} |
| \] |
\] |
| exists and are all continuous for any $i,j$ from $1$ to $m$. |
exists and are all continuous for any $i,j$ from $1$ to $m$. |
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| \subsubsection*{$C^n$ functions on an open set of $\R^m$} |
Sometimes we need to talk about continuity not globally but on some interval or region. If $f:\R^m\to\R^n$ and $U\subseteq \R^m$, then we say that $f$ is $C^n(U)$ if all the $n$-th order derivatives exist and are continuous on $U$. |
| Sometimes we need to talk about continuity not globally on $\R$, |
Usually $U$ is an open region, or some interval. Sometimes we can also speak of $C^n$ functions when range is $\C$. |
| but on some interval or open set. |
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| If $U\subseteq \R^m$ is an open set, and $f\colon U\to \R$ |
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| (or $f\colon U\to \C$) |
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| we say that $f$ is of class $C^n$ if $\partial^\alpha f$ |
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| exist and are continuous for all multi-indices $\alpha$ with $|\alpha|\le n$. |
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| See \PMlinkname{this page}{MultiIndexNotation} for the multi-index notation. |
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