I am seeking for a $C^1$ function s.t. $f(x)=0$ iff $x\in(-\infty,x_1]\cup[x_2,\infty)$. I can imagine that $f$ is something like this:
Do you know about some function like this?
Thank you so much.
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1Have you tried writing a piecewise formula, and if so, what seems not to be working? – Andrew D. Hwang Mar 30 '22 at 23:53
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2One would be $f(x)=\begin{cases}0&\text{if }x\le x_1\lor x\ge x_2\ \exp\frac1{(x-x_1)(x-x_2)}&\text{if }x_1<x<x_2\end{cases}$, which is also $C^\infty$. – Sassatelli Giulio Mar 30 '22 at 23:57
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1Related: https://math.stackexchange.com/questions/101480/are-there-other-kinds-of-bump-functions-than-e-frac1x2-1 – JackT Mar 31 '22 at 00:02
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@SassatelliGiulio perfect! Thank you really much. – Quiet_waters Mar 31 '22 at 00:38
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@JackT, great question, thank you! – Quiet_waters Mar 31 '22 at 00:38
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@AndrewD.Hwang I was with trouble to mantain the property $C^1$ because I did not know about bumps. Thank yoU! – Quiet_waters Mar 31 '22 at 00:39
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2If it matters, to ensure $C^1$ it's enough to ensure the "bump" has derivative $0$ at the endpoints $x_1$ and $x_2$, e.g., $f(x) = (x - x_1)^2(x - x_2)^2$ on $[x_1,x_2]$. – Andrew D. Hwang Mar 31 '22 at 02:01
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@AndrewD.Hwang thank you! – Quiet_waters Mar 31 '22 at 13:13