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I need to find an infinitely differentiable function from $\mathbb{R}$ to $\mathbb{R}$ which is zero for all negative values and nonzero fo all positive values.

Thank you in advance

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A standard example is $f(x)=e^{-1/x}$ if $x>0$ and $f(x)=0$ otherwise. – wj32 Oct 31 '12 at 11:22
up vote 1 down vote accepted

$$f(x)=\left\{\begin{array}{rcl} 0 &\mbox{if} & x\leq 0 \\ \exp\left(-\frac{1}{x^2}\right)&\mbox{if}&x>0\end{array}\right.$$ is a good candidate. For any $n\in\mathbb{N}$ we have: $$\frac{d^n}{dx^n}\exp\left(-1/x^2\right) = p(1/x) \exp\left(-1/x^2\right)$$ where $p$ is a polynomial, so: $$\lim_{x\to 0^+} \frac{d^n}{dx^n}\exp\left(-1/x^2\right) = \lim_{z\to +\infty} p(z)\,e^{-z^2} = 0.$$

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Is there a way to show that this function is infinitely differentiable ? – user43418 Oct 31 '12 at 11:23
    
@user43418: See en.wikipedia.org/wiki/Non-analytic_smooth_function – wj32 Oct 31 '12 at 11:26

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