The continuous functions is determined by a dense subset of X, that is, $\vert C^*(X)\vert\leq\vert C^*(D)\vert$.
The density $d(X)$ of a space $X$ is the smallest cardinality of a dense subset of $X$. To be more precise $$d(X)=\aleph_0+\min\{|D|; D\subseteq X\text{ is dense}\}$$
If $D$ is a dense subset of $X$ such that $|D|=d(x)$, then the cardinality of the set of all real-valued continuous functions is less or equal to the cardinality of the set of all real-valued functions defined on the set $D$, i.e., $\mathfrak c^{d(X)}=2^{\aleph_0 d(X)}$.
Can you take it from there?
The basic idea is almost the same as in the proof that the cardinality of $C(\mathbb R)$ is $\mathfrak c$: Cardinality of set of real continuous functions.
