Given a functional $\phi$, it is common in physics texts to see its functional derivative defined as $$\frac{\delta \phi(f)}{\delta f} = \frac{d }{d\epsilon}\phi(f + \epsilon\psi)\Big|_{\epsilon = 0}$$ where $\psi$ is a test function. However, in Folland's book on quantum field theory, he defines it as $$\frac{\delta \phi(f)}{\delta f} = \lim_{\epsilon \rightarrow 0} \frac{\phi(f + \epsilon \delta_x) - \phi(f)}{\epsilon}$$ where $\delta_x$ is the delta-function with pole at $x$. Folland does point out that the above is only a formal definition.
My first question is if $\phi$ belongs to the dual of some function space, how is it permitted to have it act on another functional (in this case, the delta-function)? Secondly, what is stopping the above definitions from being rigorous? Is there anyway to make them rigorous?