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For every continuous function $g$ on the real line and for every positive continuous function $\epsilon$ on the real line, there exists an entire function $f$ such that $|f(x)-g(x)|<\epsilon(x)$ for all real $x$ (This is due to Carleman). So if you can construct a real continuous function with your property then you can also construct an entire one. But construction of a real continuous function with this property does not seem to be difficult.