Today, I had the desire to construct a [itex]C^{\infty}[/itex] approximation to a tent function. Specifically, for any positive real number(adsbygoogle = window.adsbygoogle || []).push({}); eI want a [itex]C^{\infty}[/itex] functionfsuch that:

f(x) = 0if|x| > 1 + e

|f(x) - g(x)| < efor allx

whereg(x)is the tent function given by:

[tex]

g(x) =

\begin{cases}

0 & |x| \geq 1 \\

1 - |x| & |x| \leq 1

\end{cases}

[/tex]

I'm willing to accept on faith that such things exist, but it struck me today that I don't know how to go aboutconstructingsuch a thing, or at least proving its existence.

Given time I could probably figure it out, but I'm interested in a different problem (for which I want to use this), and I imagine this is a well-known thing.

So I guess what I'm looking for is at least a "yes" or "no" answer to the existence of such a function, but a hint as to the proof would be nice too.

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# Constructing a function!

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