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A continuous real-valued function is a continuous map whose codomain is the real line.

Write $C(\mathbb{R},\mathbb{R})$ for the set of continuous real-valued functions on the real line. Constant functions give an injection of $\mathbb{R}$ into this set. In the other direction, restriction to the rational numbers is injective. Indeed, if continuous functions $f,g$ agree at every rational and $x$ is real, choose rational numbers $q_n\to x$. Continuity givesThus restriction embeds $C(\mathbb{R},\mathbb{R})$ into ${\mathbb{R}}^{\mathbb{Q}}$, which has the same cardinality as the real-sequence space because $\mathbb{Q}$ is countable. The previous part gives the upper bound $2^{{\aleph }_0}$, while constant functions give the matching lower bound. HenceThis computes the cardinality of continuous real functions. All five sets consequently have the same cardinality. The continuity assumption is decisive for restriction to a dense subset to determine a function.