Discontinuity of a Function: Definition, Types, Examples

A function is discontinuous if we cannot sketch its graph without lifting the pen. In this article, we will study discontinuous functions with their types, examples, and a few solved problems.

At first, we recall the definition of the continuity of a function. A function f(x) is called continuous at x=a if we have lim_x→af(x) = f(a). For more details, please visit our page on “Continuity of a function: Definition, Properties, Solved Examples“.

Definition of Discontinuity of a Function

If a function f(x) is not continuous at x=a, then f(x) is said to be discontinuous at x=a. In this case, x=a is called a point of discontinuity of f(x). The function f(x) will be discontinuous at x=a if one of the following is satisfied.

1. f(a) is undefined, that is, f(x) does not have a definite value at x=a.
2. lim_x→af(x) does not exist.
3. lim_x→af(x) ≠ f(a).

Examples of Discontinuity

Below are a few examples of discontinuous functions.

1	The function f(x)=|x-1| is discontinuous at x=1.
2	f(x)=1/x is not continuous at x=0.
3	f(x)=sin(1/x) is discontinuous at x=0.
4	The quotient $\dfrac{f(x)}{g(x)}$ is discontinuous at the points where g(x)=0 but f(x)≠0.

Types of Discontinuity

There are different types of discontinuous functions. The following are the types of discontinuity of a function.

• Jump Discontinuity:

If both lim_x→a+f(x) and lim_x→a-f(x) have finite values and at least two of three quantities f(a), lim_x→a+f(x), and lim_x→a-f(x) are different, then we say that f(x) has a jump discontinuity at x=a. In this case, the difference lim_x→a+f(x) – lim_x→a-f(x) is called the height of the jump at x=a.

• Removable Discontinuity:

If lim_x→a+f(x) and lim_x→a-f(x) both exist and are equal but this is not equal to f(a), then f(x) is said to have a removable discontinuity of the first kind at x=a.

If lim_x→a+f(x) and lim_x→a-f(x) both exist and have unequal values, then f(x) is said to have a non-removable discontinuity of the first kind at x=a.

• Essential or Infinite Discontinuity:

If either lim_x→a+f(x) or lim_x→a-f(x) or both are infinite (that is, ±∞), then the function f(x) has an essential discontinuity or an infinite discontinuity at x=a.

• Oscillatory Discontinuity:

In a neighborhood of x=a, if f(x) takes values between two finite quantities infinitely often, then f(x) is said to have an oscillatory discontinuity at x=a.