In many practical situations, we encounter Gaussian distributions, for which the distribution tails are light -- in the sense that as the value increases, the corresponding probability density tends to 0 very fast. There are many theoretical explanations for the Gaussian distributions and for similar light-tail distributions. In practice, however, we often encounter heavy-tailed distributions, in which the probability density is asymptotically described, e.g., by a power law. In contrast to the light-tail distributions, there is no convincing theoretical explanation for the heavy-tailed ones. In this paper, we provide such a theoretical explanation. This explanation is based on the fact that in many applications, we approximate a continuous distribution by a discrete one. From this viewpoint, it is desirable, among all possible distributions which are consistent with our knowledge, to select a distribution for which such an approximation is the most accurate. It turns out that under reasonable condition, this requirement (of allowing the most accurate discrete approximation) leads to the desired power-law heavy-tailed distributions.