A distribution, by the definition given in Shearer and Levy, is a function $f \in D'(\mathbb{R})$ with $f: C^{\infty}_c(\mathbb{R})=D(\mathbb{R}) \rightarrow \mathbb{R}$ s.t.
- $f$ is linear
- $f$ is continuous.
A distribution is zero iff $f(\phi)=(f,\phi)=0, \ \ \forall \phi \in D(\mathbb{R})$. Given $c \in C^\infty(\mathbb{R})$ and $cf$ is defined in the image below [orange], find all values of $f$ for which $c=e^x$ satisifies $cf=0$.
Attempt:
I have tried to use the example [pink] below (also pictured) but I didn't get very far because I can't find a general function that differentiates to $c(x)e^x$. I thought integration by parts would help but that also depends on what $c(x)$ is. 
It is enough to observe that if $\phi \in \mathcal D$ then so is $e^{-x} \phi$.
Then by definition, $cf = 0$ implies $f(c \psi)=0$ for every $\psi \in \mathcal D$, and in particular, if we take any test function $\phi\in \mathcal D$, $e^{-x} \phi \in \mathcal D$ so that $$ (cf)(e^{-x}\phi) = 0 = f\big(c (e^{-x}\phi)\big) = f(\phi) $$ so $f(\phi)=0$ for all $\phi \in \mathcal D$. Therefore $f = 0$.