Consider the real-valued function *f *defined by for every real number. Note that the function returns the greatest integer less than or equal to x. The graph of *f* is portrayed below.

As can be seen from the graph, there are “jumps” at every , where is the set of all integers. The function *f* is an example of discontinuous function. Studies of continuous functions are worthy because numerous theorems in calculus (such as intermediate-value theorem and mean-value theorem) require that the functions under consideration are continuous.

**Definition**

Let f be defined on an open interval containing *c*. We say that f is ** continuous** at

*c*if

Regarding to the requirement that above, the technical proof includes three parts as follows.

**Part 1: Prove that the limit exists.**

We have to prove that there exists L ∈ such that

**Part 2: Prove that f(c) is defined. **

**Part 3: Prove that L = f(c).**

If any of the requirements is not satisfied, then we conclude that *f* is discontinuous at x = *c*.

**Example 1**

Let the function be defined by:

Is *f* continuous at x = 3?

**Answer**

Step 1: Check the existence of the limit of the function at x = 3

The left-hand limit:

The right-hand Limit:

It turns out that the one-sided limits are of equal value, that is 4. Consequently, we conclude that exists. Moreover,

Step 2: Check whether f is defined at x = 3

By the definition of *f*, the value of *f*(x) at x = 3 is defined, i.e. *f*(3) = 2.

Step 3: Check whether the limit of the function at x = 3 equals the value of the function at x = 3

From the previous steps it is easily seen that , so the conclusion is: ** f is not continuous (or discontinuous) at x = 3**. The graph of

*f*is shown in the figure below.

**Figure 1**

Notes:

The discontinuity at x = 3 in Example 1 is called removeable discontinuity. By redefining the value of *f* at x = 3, the function becomes continuous. So, for *f* to be continuous at x = 3, we have to define *f*(3) = 4.

**Example 2**

Let the function be defined by:

Is g continuous at x = 0?

**Answer**

Step 1: Check the existence of the limit of the function at x = 0

The left-hand limit:

The right-hand Limit:

It turns out that the values of the one-sided limits are not equal. Therefore, we conclude that does not exist. This indicates that one of the continuity criteria is not satisfied. So, we conclude that ** g is not continuous at x = 0**. The graph of

*g*is shown in Figure 2.

**Figure 2**

Notes:

The discontinuity at x = 0 in Example 2 is called nonremovable discontinuity. We cannot redefine the value of *g* at x = 0 to make g continuous there.

**Example 3**

Let the function be defined by:

Is *h* continuous at x = 0?

**Answer**

One of the conditions for *h* to be continuous at x = 0 is that *h* is defined at x = 0. But in this example *h*(0) is not defined. Hence, *h* is not continuous at x = 0. The graph of *h* is shown in Figure 3.

**Figure 3**

**Example 4**

Let the function be defined by:

Is *k* continuous at x = -1?

**Answer**

Step 1: Check the existence of the limit of the function at x = -1

The left-hand limit:

The right-hand limit:

Since the one-sided limits are equal, we conclude that exists. Moreover, .

Step 2: Check whether f is defined at x = -1

We conclude that k is defined at x = -1.

Step 3: Check whether the limit of the function at x = -1 equals the value of the function at x = -1

From the previous steps it is easily seen that , so the conclusion is: ** k is continuous at x = -1**. The graph of

*k*is shown in the figure below.

**Figure 4**

As Figure 4 shows, in the graph of continuous functions there are no “jumps” such as the graph of *g* (Figure 2) or the graph of *f* (Figure 1). Also, in the graph of continuous functions there are no gaps or “broken” sections such as the graph of *h* (Figure 3). The graph of the *k* function appears “continuous” without any gaps or jumps. The definition given at the beginning of this post is a formal mathematical formulation to explain this notion of continuity.