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@Somos the problem with such map is that it is not a bijection. Indeed, the representation $\sum_{i=-\infty}^\infty a_i p^i$ of $p$-adic numbers is unique, while the representation $\sum_{i=-\infty}^\infty a_i p^{-i}$ of real numbers is not.
@Lubin It seems to me that you are suggesting to plot the p-adic valuation of the function, instead of the value of the function. This would be a very rough approximation, since the p-adic valuation is discrete.