Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only take $N = 3$ distinct values it is a discrete random variable with a probability mass function.
For example, we could have $N = 3$ people with heights $150$ cm, $160$ cm, and $170$ cm, and thus the probability of any particular height occuring is $1/3$.
Now, consider the case of $X^N$ when $N = 6$ billion, i.e. the heights of the real-world human population. We now have $6$ billion distinct values for $X^N$. Although there is now a very large range of values $X^N$ can take, it is still a discrete random variable as those $6$ billion discrete values are the only values of $X^N$ that can occur.
Therefore, heights of humans is actually a discrete random variable and not a continuous random variable? Everywhere I look it says that human heights is a continuous random variable with a pdf, but it seems from the above that it is actually a discrete random variable with a pmf?