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This is an excercise 2.2 from Hormander, vol. I:

Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction $x \rightarrow e^{1/x}$ to $\mathbb{R}_+$?

The answer, provided in the book, is "No". I am trying to "cook up" appropriate test function(s) such that $ \int \phi(x)e^{1/x} \leq C\sum_{\alpha \leq k} \sup\left|\partial^{\alpha}\phi\right|$ for no $k$, and I'm not sure at all what function(s) to take. What is the appropriate function? Is there a general method to come up with just right test functions?

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Take a single nonnegative $\varphi\in D$ with support on $(1,2)$ and just consider the scaled copies $\varphi(bx)$. –  fedja Nov 2 '12 at 3:39
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I think the argument is simpler than you expect, unless I make a terrible mistake...

Just pick any test function with the property that $\phi(0) >1$ ($\phi(0)\neq 0$ is actually enough).

Since $\phi(0)>1$ there exists some $a$ so that $\phi >1$ on $[0,1]$.

Then

$$\int \phi(x) e^{\frac{1}{x}}> \int_0^a e^{\frac{1}{x}} dx = \int_\frac{1}{a}^\infty \frac{e^u}{u^2}du = \infty $$

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Yes, you make a terrible mistake: nobody said that the distribution is given by the integral formula for any functions except those whose supports stay away from $0$. However you are on the right track: create a sequence of functions converging to $0$ in $D$ such that their supports do not touch the origin but the corresponding integrals do not tend to $0$. –  fedja Nov 2 '12 at 3:36
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