題組內容

1. Consider the function f(x) = [x] - [x], where [x] denotes the smallest integer that is greater than or equal to x, and [x] denotes the greatest integer that is smaller than or equal to x. (That is, [x] = \( n \in \mathbb{Z} \) , and \( \lfloor x \rfloor = m \in \mathbb{Z} \) such that m ≤ x < m + 1.)

1.6      (3%)      True or False: Given any \( a, b \in \mathbb{R} \) such that \( a < b \), if there exists a function \( g(x) \) such that $$\int_a^b f(x)dx = \int_a^b g(x)dx, \text{ then } f(x) = g(x) \text{ for all } x \in [a, b].$$