How To Check If A Function Is Continuous

Continuity is a fundamental concept in calculus and real analysis, describing functions that have no abrupt breaks or jumps. A continuous function allows you to trace its graph without lifting your pen from the paper. Determining whether a function is continuous at a particular point or over its entire domain is crucial in many areas of mathematics, physics, and engineering. This article provides a comprehensive guide on how to check if a function is continuous, covering the theoretical basis, practical steps, and illustrative examples.

Understanding Continuity

Before diving into the methods for checking continuity, it's essential to define what continuity means mathematically. A function f(x) is said to be continuous at a point x = c if it satisfies the following three conditions:

1. f(c) is defined: The function must be defined at the point c, meaning that c is in the domain of f.

2. The limit of f(x) as x approaches c exists: The limit must exist, which means that the left-hand limit and the right-hand limit both exist and are equal. Mathematically, this is expressed as:

   lim x→c- f(x) = lim x→c+ f(x)

3. The limit of f(x) as x approaches c is equal to f(c): The value of the limit must be the same as the value of the function at c. Mathematically, this is expressed as:

   lim x→c f(x) = f(c)

If any of these three conditions are not met, the function is said to be discontinuous at x = c. There are several types of discontinuities, including:

• Removable Discontinuity: The limit exists, but it is not equal to the function's value at that point, or the function is not defined at that point. This type of discontinuity can be "removed" by redefining the function at that point.
• Jump Discontinuity: The left-hand limit and the right-hand limit both exist, but they are not equal. The function "jumps" from one value to another.
• Infinite Discontinuity: The function approaches infinity (or negative infinity) as x approaches c. This is often associated with vertical asymptotes.
• Oscillating Discontinuity: The function oscillates wildly near x = c, making it impossible to define a limit.

Steps to Check Continuity at a Point

To determine whether a function f(x) is continuous at a point x = c, follow these steps:

Step 1: Check if f(c) is Defined

The first step is to ensure that the function f(x) is defined at x = c. This means that c must be in the domain of f. If f(c) is undefined, the function is discontinuous at x = c, and you can stop here.

Example:

Consider the function f(x) = (x^2 - 1) / (x - 1). If we want to check for continuity at x = 1, we first evaluate f(1).

f(1) = (1^2 - 1) / (1 - 1) = 0 / 0

Since f(1) is undefined (division by zero), the function f(x) is discontinuous at x = 1.

Step 2: Find the Limit of f(x) as x Approaches c

Next, you need to determine if the limit of f(x) as x approaches c exists. This involves evaluating both the left-hand limit and the right-hand limit.

a. Evaluate the Left-Hand Limit:

The left-hand limit, denoted as lim x→c- f(x), is the value that f(x) approaches as x approaches c from values less than c.

b. Evaluate the Right-Hand Limit:

The right-hand limit, denoted as lim x→c+ f(x), is the value that f(x) approaches as x approaches c from values greater than c.

c. Compare the Left-Hand and Right-Hand Limits:

If the left-hand limit and the right-hand limit are equal, then the limit of f(x) as x approaches c exists and is equal to their common value. If they are not equal, the limit does not exist, and the function is discontinuous at x = c.

Example:

Let's consider the function

f(x) = { x^2, if x < 2; 3x - 2, if x ≥ 2 }

To check for continuity at x = 2, we need to evaluate the left-hand and right-hand limits:

• Left-Hand Limit:

  lim x→2- f(x) = lim x→2- x^2 = (2)^2 = 4

• Right-Hand Limit:

  lim x→2+ f(x) = lim x→2+ (3x - 2) = 3(2) - 2 = 4

Since the left-hand limit and the right-hand limit are equal (both are 4), the limit of f(x) as x approaches 2 exists and is equal to 4.

Step 3: Check if lim x→c f(x) = f(c)

Finally, you need to check if the value of the limit you found in Step 2 is equal to the value of the function at x = c, which you found in Step 1. If they are equal, the function is continuous at x = c. If they are not equal, the function is discontinuous at x = c.

Example:

Continuing with the previous example:

We found that lim x→2 f(x) = 4. Now, we need to evaluate f(2):

f(2) = 3(2) - 2 = 4

Since lim x→2 f(x) = f(2) = 4, the function f(x) is continuous at x = 2.

Checking Continuity Over an Interval

A function is said to be continuous over an interval if it is continuous at every point in that interval. To check continuity over an interval, you need to consider the following:

1. Continuity at Interior Points: The function must be continuous at every point within the interval.
2. Continuity at Endpoints: For closed intervals, you need to check continuity at the endpoints. At the left endpoint a, you need to check if lim x→a+ f(x) = f(a). At the right endpoint b, you need to check if lim x→b- f(x) = f(b).

Example:

Consider the function f(x) = √x on the interval [0, ∞).

• For any c > 0, the function is continuous because the limit as x approaches c is equal to √c, which is f(c).
• At the endpoint x = 0, we need to check if lim x→0+ √x = √0. Since lim x→0+ √x = 0 and √0 = 0, the function is continuous at x = 0.

Therefore, the function f(x) = √x is continuous on the interval [0, ∞).

Common Continuous Functions

Certain types of functions are inherently continuous over their domains:

• Polynomial Functions: Functions of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_i are constants, are continuous everywhere.
• Rational Functions: Functions of the form R(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, are continuous everywhere except where Q(x) = 0.
• Trigonometric Functions: sin(x) and cos(x) are continuous everywhere. tan(x), sec(x), csc(x), and cot(x) are continuous everywhere except at points where they are undefined (e.g., tan(x) at x = π/2 + kπ, where k is an integer).
• Exponential Functions: Functions of the form f(x) = a^x, where a > 0, are continuous everywhere.
• Logarithmic Functions: Functions of the form f(x) = log_a(x), where a > 0 and a ≠ 1, are continuous for x > 0.
• Root Functions: Functions of the form f(x) = √x are continuous for x ≥ 0.

Understanding that these functions are continuous can simplify the process of checking continuity for more complex functions that involve combinations of these basic functions.

Advanced Techniques and Considerations

The Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a crucial theorem related to continuous functions. It states that if f(x) is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.

The IVT is often used to prove the existence of solutions to equations. For example, if you want to show that the equation f(x) = 0 has a solution in the interval [a, b], you can show that f(a) and f(b) have opposite signs, and since f(x) is continuous, there must be a point c in (a, b) where f(c) = 0.

Composition of Continuous Functions

If f(x) and g(x) are continuous functions, then their composition f(g(x)) is also continuous, provided that g(x) is continuous at c and f(x) is continuous at g(c).

This property is useful when dealing with complex functions that are formed by composing simpler continuous functions. For example, h(x) = sin(x^2) is continuous everywhere because x^2 and sin(x) are both continuous everywhere.

Piecewise Functions

Piecewise functions require careful attention when checking for continuity. As demonstrated in earlier examples, you need to check the continuity at each point where the function's definition changes. This involves verifying that the left-hand limit, right-hand limit, and the function value are all equal at these transition points.

Examples of Checking Continuity

Let's go through some additional examples to illustrate the process of checking continuity:

Example 1: A Rational Function

Consider the function f(x) = (x + 3) / (x - 2). Determine where this function is continuous.

Solution:

• The function is a rational function, so it is continuous everywhere except where the denominator is zero.
• The denominator is x - 2, which is zero when x = 2.
• Therefore, f(x) is continuous for all x ≠ 2.

Example 2: A Piecewise Function

Consider the function

f(x) = { x + 1, if x < 1; x^2, if x ≥ 1 }

Check if this function is continuous at x = 1.

Solution:

1. Check if f(1) is defined:

   f(1) = (1)^2 = 1

   So, f(1) is defined.

2. Find the limit of f(x) as x approaches 1:

   • Left-Hand Limit:

     lim x→1- f(x) = lim x→1- (x + 1) = 1 + 1 = 2

   • Right-Hand Limit:

     lim x→1+ f(x) = lim x→1+ (x^2) = (1)^2 = 1

Since the left-hand limit (2) and the right-hand limit (1) are not equal, the limit of f(x) as x approaches 1 does not exist.

Therefore, the function f(x) is discontinuous at x = 1.

Example 3: A Function with a Removable Discontinuity

Consider the function f(x) = (x^2 - 4) / (x - 2). Determine if this function is continuous at x = 2.

Solution:

1. Check if f(2) is defined:

   f(2) = (2^2 - 4) / (2 - 2) = 0 / 0

   So, f(2) is undefined.

2. Find the limit of f(x) as x approaches 2:

   We can simplify the function:

   f(x) = (x^2 - 4) / (x - 2) = (x - 2)(x + 2) / (x - 2)

   For x ≠ 2, f(x) = x + 2.

   lim x→2 f(x) = lim x→2 (x + 2) = 2 + 2 = 4

Since f(2) is undefined but the limit as x approaches 2 exists, the function has a removable discontinuity at x = 2. If we redefine f(2) = 4, the function would be continuous at x = 2.

Conclusion

Checking continuity is a fundamental skill in calculus and analysis. By understanding the three conditions for continuity and following the steps outlined in this article, you can determine whether a function is continuous at a point or over an interval. Remember to pay special attention to piecewise functions, rational functions, and functions with potential removable discontinuities. Understanding the properties of common continuous functions and applying theorems like the Intermediate Value Theorem can further simplify the process. With practice, you'll become proficient at identifying and analyzing continuous functions, which is essential for solving a wide range of mathematical problems.