For a continuous distribution the probability of someone being exactly 1.78 m tall (the mean) is zero only under the assumption of infinite measurement precision.
In real-world science measurement precision is necessarily limited by instruments (or by Planck-like lower bounds). With, say, 10^-4 precision, the situation involves closed intervals like [1.7751, 1.7850], each of which has measure ≥ 0
For a continuous distribution the probability of someone being exactly 1.78 m tall (the mean) is zero only under the assumption of infinite measurement precision.
In real-world science measurement precision is necessarily limited by instruments (or by Planck-like lower bounds). With, say, 10^-4 precision, the situation involves closed intervals like [1.7751, 1.7850], each of which has measure ≥ 0