Referencing the function above, we could say that it is continuous over various intervals, such as, but that it is discontinuous over the interval since there are 3 discontinuities within that interval. For a function to be continuous over a given interval, it must be continuous at every point over that interval. Note that continuity is typically tested at a single point, or over a given interval. The function is discontinuous at point a because it is undefined it is discontinuous at point b because the limit of f(x) does not exist at that point since the left and right-handed limits are not equal it is discontinuous at point c because while the limit exists, f(5) and the limit as x approaches 5 have different values. Įach of these cases tests discontinuity at a single point, and the function below provides an example of each of these cases. f(a) is defined and the limit exists, but.A function f(x) has a discontinuity at a point x = a if any of the following is true: The formal definition of discontinuity is based on that for continuity, and requires the use of limits. These types of discontinuities are discussed below. The function on the left exhibits a jump discontinuity and the function on the right exhibits a removable discontinuity, both at x = 4. The figure below shows two functions with different types of discontinuities: Informally, a discontinuous function is one whose graph has breaks or holes a function that is discontinuous over an interval cannot be drawn/traced over that interval without the need to raise the pencil. Home / calculus / limits and continuity / discontinuity Discontinuityįunctions are classified as continuous or discontinuous.
0 Comments
Leave a Reply. |