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The closure is the set of uniformly continuous bounded functions. At first, each function $f$ in the closure is bounded, as you note, and it is uniformly continuous: If $\|f-f_n\|<\delta/3$ and $|f_n(x)-f_n(y)|<\delta/3$ whenever $|x-y|<\varepsilon$, then $||f(x)-f(y)|<\delta|$ whenever $|x-y|<\varepsilon$. On the other hand, if $f$ is bounded and uniformly continuous, then $f$ is norm-approximated by the functions $f_n(x)=n\int_x^{x+1/n}f(t)dt$, which have bounded derivatives $f_n'=n(f(x+1/n)-f(x))$.