This question seems to be posed too quickly (without substantial thinking) and has an (almost trivial) affirmative answer: We should prove that $\mathrm{Cont}(\mathbb R,\mathbb R)$ intersects each non-empty open set $U\subset \mathbb R^{\mathbb R}$. We can assume that $U$ is of basic form: $U=\prod_{r\in\mathbb R}U_r$, where for each $r\in\mathbb R$ the set $U_r$ is open in $\mathbb R$ and the set $F=\{r\in\mathbb R:U_r\ne\mathbb R\}$ is finite. Now take any (piecewise linear) continuous function $f:\mathbb R\to\mathbb R$ such that $f(r)\in U_r$ for any $r\in F$. Then $f\in \mathrm{Cont}(\mathbb R,\mathbb R)\cap U$. So, $\mathrm{Cont}(\mathbb R,\mathbb R)$ is dense in $\mathbb R^{\mathbb R}$ (even for the Tychonoff product topology of the real lines endowed with the discrete topology).