Consider the real-valued function f defined by f(x)= \lfloor x \rfloor 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 x \in \mathbb{Z}, where \mathbb{Z} 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 \lim_{x \to c} f(x) = f(c)

 

Regarding to the requirement that \lim_{x \to c} f(x) = f(c) above, the technical proof includes three parts as follows.

 

Part 1: Prove that the limit exists.

We have to prove that there exists L ∈ \mathbb{R} such that \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L

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 f : \mathbb{R} \to \mathbb{R} 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 \lim_{x \to 3} f(x) exists. Moreover, \lim_{x \to 3} f(x) = 4

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 \lim_{x \to 3} f(x) \neq f(3), 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 g : \mathbb{R} \to \mathbb{R} 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 \lim_{x \to 0} g(x) 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 h : \mathbb{R} \to \mathbb{R} 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 k : \mathbb{R} \to \mathbb{R} 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 \lim_{x \to -1} k(x) exists. Moreover, \lim_{x \to -1} k(x) = 0.

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 \lim_{x \to -1} k(x) = k(-1), 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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