Sawtooth function: Draw the graph of f(x) = x - [x]. At which points is f discontinuous? Is it left- or right-continuous at those points?
To draw the graph of the sawtooth function f(x) = x - [x], we need to understand the concept of the greatest integer function [x]. The greatest integer function, also known as the floor function, evaluates to the largest integer less than or equal to the input value.
Now, let's analyze the behavior of the sawtooth function f(x) = x - [x] to identify its points of discontinuity and determine whether it is left- or right-continuous at those points.
1. Points of Discontinuity:
The sawtooth function f(x) = x - [x] is discontinuous at any integer value of x. This discontinuity occurs because the greatest integer function [x] changes abruptly when x goes from one integer to the next. At each integer x = n, where n is an integer, the function f(x) has a jump in its graph.
For example, at x = 2.5, f(2.5) = 2.5 - [2.5] = 2.5 - 2 = 0.5, but at x = 3, f(3) = 3 - [3] = 3 - 3 = 0. These different values on either side of x = 3 indicate the discontinuity.
2. Left- or Right-Continuity:
At points of discontinuity, the sawtooth function f(x) is neither left- nor right-continuous. This means that the limit of f(x) as x approaches the discontinuity from the left side is not equal to the limit of f(x) as x approaches the discontinuity from the right side.
For example, as we approach x = 3 from the left side (x = 2.9, 2.99, 2.999, etc.), the value of f(x) gradually decreases. However, as we approach x = 3 from the right side (x = 3.1, 3.01, 3.001, etc.), the value of f(x) remains constant at 0. Therefore, f(x) is not left-continuous at x = 3.
In summary, the sawtooth function f(x) = x - [x] is discontinuous at all integer values of x, and it is neither left- nor right-continuous at those points.