Mode of a Distribution
For a discrete random variable `X`, the mode is any value `x` that maximizes the PMF `f_X(x)`.
For continuous random variables, it's a little more complicated because we can always change the PDF at isolated points without changing the probability distribution. One way to define the mode in this case is to use the concept of an essential supremum ("ess sup"). For a good explanation of ess sup, see here.