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My question is rather simple : are derivatives of bump functions still bump functions ? For example, for a bump function $u\in D(\mathbb{R}^d)$, that is,

$$u \in C^\infty|\;\text{supp}\;u\subseteq K : \mathbb{R}^d \to \mathbb{R}$$

Is this always true for all derivatives?

$$\frac{\partial u}{\partial x_j} \in D(\mathbb{R}^d)$$

I'm sure it is and couldn't imagine the contrary but I have an unexplainable doubt.

mwoua
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1 Answers1

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Yes, because, obviously, $\operatorname{supp}\partial u/\partial x_j\subset\operatorname{supp}u$.

Vladimir
  • 5,702