Cardinality of continuous maps determined on a countable dense set ($|C(X,\mathbb{R})|=2^{{\aleph }_0}(X\ne \varnothing \text{ separable})$)

Continuous real-valued functions on a space with a countable dense subset are determined by their restrictions to that subset. The equality set of two such functions is closed and contains the dense subset. Hence there are at most $|\mathbb{R}{|}^{{\aleph }_0}=2^{{\aleph }_0}$ such functions. If the domain is nonempty, distinct constant functions give the matching lower bound.