How can the direction of the electric field of a negative point charge be going in towards itself (radially), when the charge itself is *generating* the electric field? Is it generated at infinity then? Because if it's generated by the charge then it cannot go further in towards itself?

In case of positive charge, it is understandable that the field is being generated at the charge (at it's surface to be precise) and is *going outwards* radially.

[![negative point charge](https://i.stack.imgur.com/aHBsj.gif)](https://i.stack.imgur.com/aHBsj.gif)