What’s a discontinuity? Any point at which a function fails to be *continuous* is called a **discontinuity**. In fact, there are various types of discontinuities, which we hope to explain in this review article.

## Points of Discontinuity

The definition of discontinuity is very simple. A function is **discontinuous** at a point *x* = *a* if the function is *not* continuous at *a*.

So let’s begin by reviewing the definition of *continuous*.

A function *f* is **continuous** at a point *x* = *a* if the following limit equation is true.

Think of this equation as a set of three conditions.

- The limit must exist. That is, for some finite number
*L*. - The function value must exist. In other words,
*f*(*a*) exists. - The limit must agree with the function value. So, the number
*L*that you get by taking the limit should be the same value as*f*(*a*).

When one or more of these conditions fails, then the function has a discontinuity at *x* = *a*, by definition.

The first condition, that the limit must exist, is especially interesting. A limit may fail to exist for a variety of reasons. If the limit as *x* → *a* does not exist, then we can say that the function has a **non-removable discontinuity** at *x* = *a*. Then, depending on how the limit failed to exist, we classify the point further as a *jump*, *infinite*, or *infinite oscillation* discontinuity.

## Jump Discontinuities

One way in which a limit may fail to exist at a point *x* = *a* is if the left hand limit does not match the right hand limit.

In the graph shown below, there seems to be a “mismatch.” As *x* approaches 1 from the left, that part of the graph seems to land on *y* = -1. On the other hand, as *x* approaches 1 from the right, the values of *y* seem to get closer and closer to *y* = 3.

In limit notation, we would write:

Because the left and right limits do not agree, the limit of *f*(*x*) as *x* → 1 *does not exist*.

Therefore, by definition, the function *f* is discontinuous at *x* = 1. This kind of discontinuity is known as a **jump** (for obvious reasons).

## Infinite Discontinuities (Vertical Asymptotes)

In some functions, the values of the function approach ∞ or -∞ as *x* approaches some finite number *a*. In this case, we say that the function has an **infinite discontinuity** or **vertical asymptote** at *x* = *a*.

You might think of an infinite discontinuity as an extreme case of jump discontinuity.

Typically, you’ll find this behavior anywhere there is a division of the form *nonzero over zero*.

For instance 3*x*/(*x* + 12) has an infinite discontinuity at *x* = -12, because that is where the denominator becomes 0 while the numerator is nonzero (3 × -12 = -36).

### Example

Locate and classify the discontinuities of *f*(*x*) = tan *x* on the interval [-2π, 2π].

#### Solution

Note that tan *x* = (sin *x*)/(cos *x*), and the denominator, cos *x*, is equal to zero at all points of the form π/2 + *n*π. There are four *x*-values within the interval [-2π, 2π] that qualify: *x* = -3π/2, -π/2, π/2 and 3π/2. Plugging each one into the numerator, sin *x*, we verify that the top is nonzero when the bottom is zero. Thus, *f* has an infinite discontinuity at four points: *x* = -3π/2, -π/2, π/2 and 3π/2 within the interval [-2π, 2π]

The graph can help us to visualize what’s going on near those point.

## Infinite Oscillation Discontinuities

Most of the functions that you meet are fairly boring and predictable. Even at a jump or infinite discontinuity, you can say something about how the values of the function behave. However if your function has an **infinite oscillation discontinuity** at a point *x* = *a*, then things get wild!

Consider the graph of *f*(*x*) = sin(1/*x*).

Approaching the origin, the curve goes up and down more and more often. In fact as *x* → 0, from either the left or right, we can see how the oscillations get completely out of hand! It’s almost as if the curve wants to make sure it hits *every* point in the range between -1 and 1 at *x* = 0.

No matter how far we zoom in on the graph, it just gets wilder and wilder.

The reason for this strange behavior has to do with the fact that sin *x* itself is periodic. Then replacing *x* by 1/*x* in the argument has the effect of taking all those infinitely many periodic waves of the sine function as *x* → ∞ and *squeezing* them next to the origin instead.

At any rate, since there is no single value of *y* to which the curve seems to be heading as *x* → 0, the limit does not exist at *x* = 0.

Therefore, *f*(*x*) = sin(1/*x*) has a discontinuity at *x* = 0, of the infinite oscillation variety.

## Removable Discontinuities

The last category of discontinuity is different from the rest. In the previous cases, the limit did not exist. In other words, condition 1 of the definition of continuity failed. Next we’ll discuss what happens if condition 1 holds (the limit exists), but either condition 2 or 3 fail.

A function *f* has a **removable discontinuity** at *x* = *a* if the limit of *f*(*x*) as *x* → *a* exists, but either *f*(*a*) does not exist, or the value of *f*(*a*) is not equal to the limiting value.

If the limit exists, but *f*(*a*) does not, then we might visualize the graph of *f* as having a “hole” at *x* = *a*. You might imagine what happens if you filled in that hole. If it really is a *removable* discontinuity, then filling in the hole results in a continuous graph!

Let’s take a look at the graph below.

There is a hole at *x* = -2. In fact, the graph would be continuous at that point if the hole at (-2, 2) were filled in. That’s the clue that we’re dealing with a *removable *discontinuity at *x* = -2.

There is one more case to consider. Suppose both conditions 1 and 2 hold for a function at a given point, but condition 3 fails. That is, the limit exists, the function value exists, but they are *different* values.

This situation happens in the graph shown below.

Focus at what happens near *x* = 2.

There is a well-defined limit value. Because the curve approaches *y* = 3 from the left and the right, the limit is equal to 3.

But *f*(2) = 1 (based on the location of the dot). Since the value of *f* is not the same as the limiting value here, we can say that *f* has a *removable discontinuity* at *x* = 2.

### Example

Identify where the function has a removable discontinuity and determine the value of the function that would make it continuous at that point.

#### Solution

Look to the denominator. We know that *f* will be undefined whenever the bottom of the fraction is equal to 0. That happens when 2*x* – 4 = 0, or after solving for *x*, we find *x* = 2.

So we already know there is some kind of discontinuity at *x* = 2 (because *f*(2) does not exist — see condition 2 of the definition of continuity). But is it *removable*? That takes finding a limit.

Because the limit value exists, we now know for sure that there is a removable discontinuity at *x* = 2.

Moreover, the limit itself tells us what the value should be to make *f* continuous: -5/2.

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