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| Let $f:\R\to\R$ a function. We say that $f$ is of class $C^1$ if $f'$ exists and it's continuous. |
Let $f:\R\to\R$ 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 foloowing relationship among these classes: |
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| 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 |
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| C^\infty = \bigcap_{n=0}^\infty C^n. |
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| 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, $f\in C^2$ means that |
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| \frac{\partial^2 f}{\partial x^2},\frac{\partial^2 f}{\partial x\partial y},\frac{\partial^2 f}{\partial y^2} |
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| exists and are all continuous. |
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