The terms “discrete” and “continuous” are not related to cardinality, but to topology. In particular, you can have a discrete topology on a space of arbitrary cardinality.
Note also that “discrete” is a property of a set in a topological space, while “continuous” is a property of functions between topological spaces. Therefore they are not even really opposites of each other.
A subset of a topological space is discrete if its subspace topology is the discrete topology. This is the case if all the points of the set are isolated, that is, for any point of the set you can find a neighbourhood that contains no other point of the set.
Now the natural opposite of this would be a set that has no isolated points at all. It might make sense to call such sets continuous, but I'm not aware of that being done.
The common use of “continuous” is in respect to functions. In intuitive terms, a function is continuous if the image of points that are close together still stay close together, that is, you can stay close in the image by staying sufficiently close in the domain. The formal definition is that a function is continuous if the preimages of open sets are open.
Conversely you can call a function discrete if its image is discrete. In that case, the only way for the image of a point to stay close is to not move at all. Note that a function can be continuous and discrete at the same time. One obvious example are constant functions. But also functions on a discrete domain are always continuous.