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I want to drift a $C^1$ function $g(x):\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ to reach $"0"$ smoothly on some interval $[x_{1},x_{2}]$ and then drift it again to continue it's values on $(x_{2},\infty)$. I've found something about partitions of unity and bump functions but i'm a bit confused. Can i multiply $g(x)$ by the bump function that is $0$ on $[x_{1},x_{2}]$ and 1 outside it with just a small frame $(x_{1}-\epsilon,x_{1})$ in which it can fall from 1 to zero?

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Yes, you can do exactly that. It is the nice thing about bump functions.

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