 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?

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?

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?

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?

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.

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