Okay, I think I understand what you're getting at. But if you've taken a set of points from a continuous set (like an interval on the reals) and you can put those in bijection with a discrete set, then by definition your subset of the continuous set isn't continuous. It must be discrete.
More succinctly, you actually can't draw an isomorphism between discrete and continuous spaces without losing information from the continuous space.
Here's the thing, your description of the continuous set is already discretized. If we say an interval 4-6 we have captured the continuous space using only two numbers.
I know that this is a silly argument in some sense but this is something that you do naturally that you don't even think about it.
Do you see what I'm getting at? You capture extrema in the discrete space and the interpolate in the smooth space to recreate the smooth curve.
More succinctly, you actually can't draw an isomorphism between discrete and continuous spaces without losing information from the continuous space.