# Method of images / charge density uniqueness

Say I'd like to find the potential created with a conducting sphere and an external point charge. When using the method of images to find the potential, we know that that potential is unique for the region of interest (outside the sphere), as long as it is defined on all boundaries and as long as the charge density is specified in the region.

If I wanted to find the surface charge density of the conductor, using the potential $$V$$ arrived at via the method of images, I can do so with

$$\sigma=\frac{1}{4\pi}E_r = -\frac{1}{4\pi}\frac{\partial V}{\partial r}.$$

From a mathematical point of view, this makes sense. But how can a collection of point charges recreate a continuous charge distribution? The method of images emphasizes the uniqueness of the solutions, and I think that because $$V$$ is uniquely determined then $$\rho$$ must be as well. But I cannot wrap my head around how 3 point charges could create a unique, continuous charge density. I would think it should be some collection of delta functions.

But how can a collection of point charges recreate a continuous charge distribution? The method of images emphasizes the uniqueness of the solutions, and I think that because $$V$$ is uniquely determined then $$\rho$$ must be as well.