Determine Whether The Function Is Continuous

Functions are the backbone of mathematical analysis, and understanding their properties is crucial for various applications. One of the most fundamental properties of a function is its continuity. A continuous function is one whose graph can be drawn without lifting the pen from the paper. In other words, there are no abrupt jumps, breaks, or holes in the graph. This article provides a comprehensive guide on how to determine whether a function is continuous, covering the necessary definitions, theorems, and examples.

Definition of Continuity

At its core, continuity is defined at a single point. A function f(x) is said to be continuous at a point x = c if the following three conditions are met:

f(c) is defined: The function must be defined at the point c. This means that when x = c is plugged into the function, it yields a real number.
The limit of f(x) as x approaches c exists: This implies that as x gets arbitrarily close to c from both the left and the right, the values of f(x) approach a specific value. Mathematically, this is represented as:
```
lim x→c f(x) exists
```
The limit of f(x) as x approaches c is equal to f(c): This condition ensures that the value the function approaches as x nears c is the same as the value of the function at c. Symbolically:
```
lim x→c f(x) = f(c)
```

If any of these conditions are not satisfied, the function is said to be discontinuous at x = c.

A function is said to be continuous on an interval if it is continuous at every point in that interval.

Types of Discontinuities

Understanding the types of discontinuities helps in diagnosing why a function fails to be continuous. There are three primary types of discontinuities:

Removable Discontinuity: This occurs when the limit of f(x) as x approaches c exists, but it is not equal to f(c), or f(c) is undefined. This type of discontinuity can be "removed" by redefining the function at that point to be equal to the limit.
- Example:
```
f(x) = (x^2 - 4) / (x - 2)
```
  At x = 2, f(x) is undefined because the denominator is zero. However,
```
lim x→2 (x^2 - 4) / (x - 2) = lim x→2 (x + 2) = 4
```
  So, we can redefine f(x) as:
```
f(x) = { (x^2 - 4) / (x - 2), x ≠ 2
       { 4, x = 2
```
  This new function is continuous at x = 2.
Jump Discontinuity: This occurs when the left-hand limit and the right-hand limit at x = c both exist, but they are not equal. The function "jumps" from one value to another at this point.
- Example:
```
f(x) = { 1, x < 0
       { 2, x ≥ 0
```
  Here, lim x→0- f(x) = 1 and lim x→0+ f(x) = 2. Since the left and right limits are different, there is a jump discontinuity at x = 0.
Infinite Discontinuity: This occurs when the function approaches infinity (or negative infinity) as x approaches c. This is often associated with vertical asymptotes.
- Example:
```
f(x) = 1 / x
```
  As x approaches 0, f(x) approaches infinity (or negative infinity depending on the direction). Thus, there is an infinite discontinuity at x = 0.

Theorems on Continuity

Several theorems help in determining the continuity of functions more efficiently. These theorems allow us to build up complex continuous functions from simpler ones.

Continuity of Polynomials: All polynomial functions are continuous everywhere. A polynomial function is a function of the form:
```
f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
```
where a_n, a_{n-1}, ..., a_1, a_0 are constants and n is a non-negative integer.
Continuity of Rational Functions: A rational function (a function of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials) is continuous everywhere except where the denominator q(x) = 0.
Continuity of Trigonometric Functions: The sine and cosine functions are continuous everywhere. The other trigonometric functions (tan, cot, sec, csc) are continuous everywhere in their domains.
Continuity of Exponential and Logarithmic Functions: Exponential functions (e.g., f(x) = a^x) and logarithmic functions (e.g., f(x) = log_a(x), for x > 0) are continuous in their domains.
Continuity of Root Functions: The nth root function f(x) = x^(1/n) is continuous for all x if n is odd, and for x ≥ 0 if n is even.
Composition of Continuous Functions: If f(x) is continuous at x = c and g(x) is continuous at f(c), then the composite function g(f(x)) is continuous at x = c.
Arithmetic Operations on Continuous Functions: If f(x) and g(x) are continuous at x = c, then the following functions are also continuous at x = c:
- f(x) + g(x)
- f(x) - g(x)
- f(x) * g(x)
- f(x) / g(x), provided g(c) ≠ 0

Steps to Determine Continuity

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

Check if f(c) is Defined: Evaluate f(c). If f(c) is not a real number, then f(x) is discontinuous at x = c.
Compute the Limit as x approaches c: Calculate the limit:
```
lim x→c f(x)
```
This often involves finding the left-hand limit (as x approaches c from the left) and the right-hand limit (as x approaches c from the right). If these limits are not equal, the limit does not exist, and f(x) is discontinuous at x = c.
Compare the Limit to f(c): If the limit exists, check if:
```
lim x→c f(x) = f(c)
```
If this condition holds, then f(x) is continuous at x = c. Otherwise, f(x) is discontinuous at x = c.

Example 1: Piecewise Function

Consider the piecewise function:

f(x) = { x^2, x ≤ 1
       { 2x - 1, x > 1

Determine if f(x) is continuous at x = 1.

f(1) is Defined: f(1) = (1)^2 = 1.
Compute the Limit as x approaches 1:
- Left-hand limit:
```
lim x→1- f(x) = lim x→1- x^2 = 1^2 = 1
```
- Right-hand limit:
```
lim x→1+ f(x) = lim x→1+ (2x - 1) = 2(1) - 1 = 1
```
Since the left-hand limit and right-hand limit are equal, the limit exists and lim x→1 f(x) = 1.
Compare the Limit to f(1):
```
lim x→1 f(x) = 1 = f(1)
```

Since all three conditions are met, f(x) is continuous at x = 1.

Example 2: Rational Function

Consider the rational function:

f(x) = (x + 3) / (x - 2)

Determine the points where f(x) is continuous.

f(x) is a rational function, so it is continuous everywhere except where the denominator is zero.
The denominator is x - 2. Setting x - 2 = 0, we find x = 2.

Therefore, f(x) is continuous everywhere except at x = 2. At x = 2, there is an infinite discontinuity.

Example 3: Function with Removable Discontinuity

Consider the function:

f(x) = (x^2 - 9) / (x - 3)

Determine if f(x) is continuous at x = 3.

f(3) is Undefined: At x = 3, the denominator is zero, so f(3) is undefined.

Compute the Limit as x approaches 3:

lim x→3 (x^2 - 9) / (x - 3) = lim x→3 (x + 3)(x - 3) / (x - 3) = lim x→3 (x + 3) = 6

The limit exists and is equal to 6.

Compare the Limit to f(3):

Since f(3) is undefined, f(x) is discontinuous at x = 3. However, this is a removable discontinuity. We can redefine f(x) as:
```
f(x) = { (x^2 - 9) / (x - 3), x ≠ 3
       { 6, x = 3
```
This new function is continuous at x = 3.

Advanced Techniques and Considerations

Using Derivatives: If a function is differentiable at a point, it is also continuous at that point. However, the converse is not necessarily true. A function can be continuous but not differentiable (e.g., f(x) = |x| is continuous at x = 0 but not differentiable).
Intermediate Value Theorem (IVT): The IVT 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. This theorem is useful for proving the existence of roots of equations.
Extreme Value Theorem (EVT): The EVT states that if f(x) is continuous on a closed interval [a, b], then f(x) must attain a maximum value and a minimum value on that interval.
Uniform Continuity: A function f(x) is uniformly continuous on an interval I if for every ε > 0, there exists a δ > 0 such that for all x, y ∈ I, if |x - y| < δ, then |f(x) - f(y)| < ε. Uniform continuity is a stronger condition than pointwise continuity.
Continuity in Higher Dimensions: The concept of continuity extends to functions of several variables. A function f(x, y) is continuous at a point (a, b) if for every ε > 0, there exists a δ > 0 such that if √( (x - a)^2 + (y - b)^2 ) < δ, then |f(x, y) - f(a, b)| < ε.

Applications of Continuity

Continuity is a fundamental concept in mathematics with numerous applications in various fields:

Calculus: Continuity is essential for defining derivatives and integrals. The derivative of a function is defined as the limit of the difference quotient, and this limit only exists if the function is continuous.
Physics: Many physical phenomena are modeled using continuous functions. For example, the motion of an object, the temperature distribution in a room, and the flow of a fluid are often described by continuous functions.
Engineering: Engineers use continuous functions to design and analyze structures, circuits, and control systems. Continuity ensures that small changes in the input lead to small changes in the output.
Economics: Economists use continuous functions to model supply and demand, utility functions, and production functions. Continuity allows for smooth transitions and predictable behavior.
Computer Graphics: Continuous functions are used to create smooth curves and surfaces in computer graphics. Spline curves, Bézier curves, and NURBS are examples of continuous functions used in computer-aided design (CAD) and animation.

Common Mistakes to Avoid

Assuming Continuity: Do not assume a function is continuous without verifying the conditions for continuity. Always check if f(c) is defined, the limit exists, and the limit equals f(c).
Ignoring Piecewise Functions: When dealing with piecewise functions, pay close attention to the points where the function definition changes. These are the points where discontinuity is most likely to occur.
Confusing Limit Existence with Continuity: The existence of a limit does not guarantee continuity. The limit must also be equal to the value of the function at that point.
Forgetting to Check One-Sided Limits: When determining the existence of a limit, ensure that both the left-hand limit and the right-hand limit exist and are equal.
Overlooking Domain Restrictions: Be aware of domain restrictions, such as division by zero or the logarithm of a non-positive number. These restrictions can lead to discontinuities.

Conclusion

Determining whether a function is continuous is a critical skill in mathematical analysis. By understanding the definition of continuity, the types of discontinuities, and the theorems on continuity, one can effectively analyze the behavior of functions and their properties. The step-by-step approach, along with practical examples, provides a solid foundation for mastering this essential concept. Whether in calculus, physics, engineering, or economics, the principles of continuity are fundamental to understanding and modeling the world around us.