I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition -
$$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(x_i)|$$
Now, I understand that x represents partitions of the interval in which the function f is defined. But, I am confused how can the supremum be taken. I mean, essentially, it will summate the modulus of the difference between two real numbers, which will be a real number, like, say c. Why will a supremum be required?
(EDIT: sup moved outside the summation based on the helpful responses below!)