# Continuous limit of discrete position basis

Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$.
For instance how does quantities such as $\lvert i\rangle$ with $i=xa$ maps to $\lvert x \rangle$? Or the orthogonality relations $\langle i \lvert j\rangle=\delta_{ij}$ and $\langle x \lvert x'\rangle=\delta(x-x')$?

• – Qmechanic Feb 2 '18 at 18:16

$$f:\mathbb N \rightarrow \mathbb R$$