There exists the following relation for the Besselfunctions $j_{l}(\alpha\,x)$, $j_{l}(\beta\,x)$ with $\alpha$, $\beta$ $\in \mathbb{R}$ and $x\in \mathbb{R}$
$\int_{0}^{\infty}dx \, x^{2}\, j_{l}(\alpha\,x)\, j_{l}(\beta\,x) = \frac{\pi}{\beta}\, \delta(\alpha^{2}-\beta^{2}). $
Now, assume $\alpha_{n} , \beta_{n} \in \mathbb{Z}$ , i.e. are discrete numbers, then the above equation can be rewritten in terms of
$\int_{0}^{\infty}dx \, x^{2}\, j_{l}(\alpha_{n}\,x)\, j_{l}(\beta_{n}\,x) = \int_{0}^{\infty} \,dy \, dz \, \delta(y-\alpha_{n}) \, \delta(z-\beta_{n}) dx \, x^{2}\, j_{l}(y\,x)\, j_{l}(z\,x)$ $= \int_{0}^{\infty} \,dy \, dz \, \delta(y-\alpha_{n}) \, \delta(z-\beta_{n}) \frac{\pi}{z}\, \delta(y^{2}-z^{2})$ $=\frac{\pi}{\beta_{n}}\, \delta(\alpha_{n}^{2}-\beta_{n}^{2}). $
Question: I don't understand how the Delta Distribution which is supposed to be continuous can be understood in terms of discrete variables, since the Dirac Delta within the theory of generalized functions is always used with some test function which is integrated over which is not possible in this case.