To understand discontinuity, you should know what is continuity of a function. If a function doesn't have any anomalous point or breaking point, it means it is a continuous function. However, not all functions are continuous, we call them discontinuous function. A discontinuous function is a function that changes its behaviour after some time. One of the best examples of discontinuous function is the piecewise function. A piecewise function is a type of function that is defined by many sub-functions with a certain interval. In simple words, a piecewise function contains functions of its own with its own domain. These sub-functions are the indication that there are some discontinuity points between them and from those points, the function gives a different output for different inputs.

One thing is clear, piecewise functions are discontinuous function but which type of discontinuity? There are three types of discontinuity and they are:

  • Point/Removal discontinuity
  • Jump discontinuity
  • Asymptotic discontinuity

Piecewise functions are mostly jumped discontinuity function. These types of discontinuity are very detailed and deserve their own resources and that is why, in this lecture, we will discuss jump discontinuity.

What is Jump Discontinuity?

Imagine a graph of a function that has two sub-functions which are:

f(x) = \left\{\begin{matrix} { (x - 1) }^{ 0.5 } + 2 \quad for \quad 1 \leq x < 2 \\ \frac { 2 }{ { (x - 1) }^{ 2 } } \quad for \quad x \geq 2 \end{matrix}\right

Let's make graph of the above piecewise function.

A piecewise function which tells about jump discontinuity

Notice something? At point x = 2, there is a jump. The first sub-function (i.e. { (x - 1) }^{ 0.5 } + 2) ended before point x = 2 and from the same point, the second sub-function (i.e. \frac { 2 }{ { (x - 1) }^{ 2 } }) starts and goes till infinity. That means that at point x = 2, there is a jump in sub-functions. In conclusion, if a function behaves differently not just on a specific point but it has its own domain, we call it a jump discontinuity.

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How To Check Jump Discontinuity?

The best way to check jump discontinuity is to apply limits on the left and right sides of the limits. If they both are equal to each other that means there is no jump discontinuity, however, if they are not equal that indicates jump discontinuity. For example, we have a piecewise function:

f(x) = \left\{\begin{matrix} { x }^{ 2 } \quad if \quad x < 2 \\ 1 \quad if \quad x \geq 2 \end{matrix}\right

If the left and right sides of the limits at x = a exist, and they are finite but not equal, then at x = a there is a jump discontinuity or a step discontinuity.

\lim_{ x \rightarrow { a }^{ - } } f( x ) \neq \lim_{ x \rightarrow { a }^{ + } } f( x )

\lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4 \qquad \lim_{ x \rightarrow { 2 }^{ + } } 1 = 1

\lim_{ x \rightarrow { 2 }^{ - } } f( x ) \neq \lim_{ x \rightarrow { 2 }^{ + } } f( x )

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.