
The figure below shows the graph of the function s, where x € R and s(x) = {1 if x > 0; 0 if x = 0; 1 if x < 0. This function is called step function. The step function G , partially graphed below maps each real number to x into the greatest integer that is less than or equal to x. [] means step integer function. For instance, G maps 2 into 2 because the greatest integer less than or equal to 2 is 2 itself. But G also maps 2 1/2 into 2, since the greatest integer less than or equal to 2 1/2 is 2. To denote this, we use the notation G = x > [x] or we write G(x) = [x], where the symbol [x] is read the "the greatest integer that is less than or equal to x." Thus, [2] = 2, [2 1/2]= 2, [2.69] = 2, but [2 1/2] = 3, [0] = 0, [1 1/3] = 1, [1/5] = 1, [2.99] = 2, [1] = 1, [4 3/4] = 4, [3 1/10] = 4, [5.1] = 6 Definition: Let x € R, A function f(x) = [x] where f(x) = n (an integer) such that n<=x< n+1 is called a step or greatest integer function. Thus for all x such that 0<=x<1, f(x) = 0, 1<=x<2, f(x) = 1 2< = x<3, f(x) = 2 ............ ........... .............. ........... 1<=x<0 f(x) = 1 2<=x<1 f(x) = 2 ............. ............. ............. .............. Domain of f = R, Range of f = set of integers Example: Draw the graph of the following function y = [x] in 2<=x<4 Solution: y = [x]. Here the value of y is the integer value of x. Some points of the graph are When x is 2<=x<1, y is 2 When x is 1<=x<1, y is 1 When x is 0<=x<1, y is 0 When x is 1<=x<1, y is 1 When x is 2<=x<1, y is 2 When x is 3<=x<1, y is 3 i.e f(x) = { 2 when x € [2,1] 1 when x € [1, 0] 0 when x € [0,1] 1 when x € [1,2] and so on. The graph is as shown below. Clearly the function has jumps at the points (1,2), (0,1), (1,0), (2,1) etc. In other words, the given function is discontinuous at each integral value of x.
Directions: Answer the following.
