Even though there is a non-denumerable number of real numbers, there is only a denumerable number of them we can represent in a finitary logic. The real numbers which we can represent can be set to a one to one correspondence to the rational numbers, or even just the natural numbers.
But we have an intuitive knowledge of infinity and continuity that goes beyond discrete finitary intellect.
There are multiple such dualities.
Discrete — Continuous
Particle — Wave
Intellect — Intuition
Digital — Analog
Opaque — Ambiguous (???) Added 9/17/2013
Each of these has its place.
I have talked to those who think quantum mechanics indicates everything comes in discrete quanta. It is true that the energy of a photon of a specific frequency has a fixed energy (E = h v), but the frequency (v) can vary continuously.