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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Constructing a function!

**Physics Forums | Science Articles, Homework Help, Discussion**