Xn (n E lP') .
With notations as in the theorem, the integer n is denoted [x] and the function IR --+ Z defined by x 1-+ [x] is called the bracket function (or the greatest integer function , since [x] is the largest integer that is ::; x ). Exercises 1. 3] and [-2]. 2. Sketch the graph of each of the following functions f: IR (i) f(x) = [x]. } (ii) f(x) = [2x]. (iii) f(x) = [-x + 3]. --+ IR . 3. 24. Infer a general principle about 'rounding off'. 20 2. First Properties of ffi. 4. 1. Theorem. If x and yare real numbers such that x < y, then there exists a rational number r such that x < r < y .
A First Course in Real Analysis (Undergraduate Texts in Mathematics) by Sterling K. Berberian