Here is something that I found contradictory (or appears to be contradictory).
I attached as a LaTeX to make it more readable.
I do not understand what you wrote. I never studied topology. (Is topology a generalization of real analysis?)
Sometime ago I was reading something on topology and I remember they defined countinous in terms of metric spaces. It is the same thing d(x,x_0) insteand of |x-x_0|. Which seems to confirm with the "well-defined" definition.