There is a natural continuous linear surjection $C_b(\mathbb{R}) \to \ell_\infty(\mathbb{Z})$ defined by restricting the function $f \in C_b(\mathbb{R})$ to the subdomain $\mathbb{Z}\subseteq \mathbb{R}$. The space $\ell_\infty(\mathbb{Z})$ is well-known to be non-separable, and a non-separable topological space cannot be the continuous image of a separable one.

There is a natural continuous linear surjection $C_b(\mathbb{R}) \to \ell_\infty(\mathbb{Z})$ defined by restricting the function $f \in C_b(\mathbb{R})$ to the subdomain $\mathbb{Z}\subseteq \mathbb{R}$. The space $\ell_\infty(\mathbb{Z})$ is well-known to be non-separable, and a non-separable topological space cannot be the continuous image of a separable one.

To look at the same argument in a different way, define $X$ to be the closed linear subspace of $C_b(\mathbb{R})$ consisting of functions which are affine on each of the intervals $[n,n+1]$. Then $X$ is isometrically isomorphic to $\ell_\infty(\mathbb{Z})$ via the restriction $f \mapsto f|_{\mathbb{Z}}$.