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... a continuous differential map $\dfrac{d}{dx} : C^k(\mathbb R)\rightarrow C^{k-1}(\mathbb R)$ ...

I was wondering why a differential map could from $C^k(\mathbb R)$ to $C^{k-1}(\mathbb R)$, rather than from $C^{k-1}(\mathbb R)$ to $C^{k}(\mathbb R)$?

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If you have a function that is assumed to be $k$-times continuously differentiable, then all you know about its derivative is that it has $k-1$ continuous derivatives, and hence is in $C^{k-1}(\mathbb{R})$.

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