Counterexample to Liouville in reals

Question

I know if $f: \mathbb{C} \rightarrow \mathbb{C}$ is bounded and holomorphic, then it is just the constant function. My question is for $f: \mathbb{R} \rightarrow \mathbb{R}$ what would be a good example where this fails. I know trig functions are bounded and have infinitely many derivatives so would $f(x):= \sin x$ work where $f: \mathbb{R} \rightarrow \mathbb{R}$? thanks in advance.