Please correct me if I am wrong, but I understand the Pontryagin Dual for $\mathbb{Z}/n\mathbb{Z}$ and the circle in the complex plane as a set of representations for $\mathbb{Z}/n\mathbb{Z}$ defined by the continuous homomorphisms $$ \phi_k:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{C}\\ 1\mapsto e^{\frac{2\pi i k}{n}}\\ k\in\{0,1,....,n-1\} $$ Giving us $n$ different characters for $\mathbb{Z}/n\mathbb{Z}$ in $S^1$.
My notes then move onto all of $\mathbb{Z}$, seeming to suggest that the map is defined similarly, as $1$ being mapped to a point, and the rest of $\mathbb{Z}$ following as $1$ is the generator.
Assuming the above is correct, how do we extend this to $\mathbb{R}$? I understand how to get there once we have representations for points in $\mathbb{Q}$, since it follows from continuity of the map, but how do we extend to $\mathbb{Q}$ in the first place? Something along the lines of $$ \phi_k(m/n)=e^{\frac{m}{j}*\frac{2\pi i}{k}} $$ would seem obvious, but is there a good way to justify this?
Thanks!