What is the exact meaning of $C^{\infty} (\mathbb{R})$ ? what is application i mean why we used this symbol?

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What is the exact meaning of $C^{\infty} (\mathbb{R})$ ? what is application i mean why we used this symbol ?

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The expression $C^\infty(\mathbb{R})$ stands for the set of all functions from $\mathbb R$ into itself such that all derivative ($f',f'',f''',\ldots$) exist. It is used when we want to talk about these functions.