Problem 38 - Calculus of a Single Variable

Question

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=x^{2}-4 x+4 $$

Step-by-Step Solution

Answer

The function \(f(x) = x^{2} - 4x + 4\) is continuous for all x-values, hence it does not have any points of discontinuity, whether removable or non-removable.

Step 1: Identify the Type of Function

The provided function, \(f(x) = x^{2} - 4x + 4\), is a quadratic equation, which is a polynomial of degree 2. It's important to note that polynomial functions are continuous everywhere, i.e., for all real numbers.

Step 2: Identify Points of Discontinuity

As polynomial functions are continuous everywhere, there are no x-values for which the given function \(f(x)\) is not continuous. Therefore, the function does not have any points of discontinuity.

Step 3: Classify Discontinuities

As no points of discontinuity exist for the function, there are no discontinuities to be classified as removable or non-removable.

Key Concepts

Removable Discontinuities

A removable discontinuity is a point on the graph of a function where the function is not defined, or where the limit does not match the function's value, but which can be "fixed" by redefining the function at that point. These typically occur in rational functions where a factor in the denominator cancels out with a factor in the numerator.

For example, consider a function defined as \[ g(x) = \frac{x^2 - 1}{x - 1} \]If you evaluate this function at \( x = 1 \), you'll notice that both the numerator and denominator become zero, indicating an undefined value. However, this undefined point is removable because if you factor and simplify, \[ g(x) = (x + 1) \]the discontinuity at \( x = 1 \) can be "removed" by redefining \( g(1) = 2 \). This adjustment will make the function continuous at \( x = 1 \).

This means that for functions like polynomials, which are inherently continuous, we typically don't find removable discontinuities unless the function is altered to create one. In the case of \( f(x) = x^2 - 4x + 4 \), there are no alterations or divisions by zero, indicating no removable discontinuities are present.

Nonremovable Discontinuities

Nonremovable discontinuities, unlike removable ones, cannot be fixed by simply redefining a function at a point. These occur in four common forms: infinite, jump, essential, and endpoint discontinuities.

- **Infinite Discontinuities:** These occur where a function approaches infinity at a vertical asymptote.
- **Jump Discontinuities:** These are observed when there is a sudden leap in function values.
- **Essential Discontinuities:** These are neither removable nor can they be corrected with a simple redefinition of a single point.

Polynomials such as \( f(x) = x^2 - 4x + 4 \) inherently possess none of these types of discontinuities because polynomial functions are continuous everywhere. Unlike rational, trigonometric, or piecewise functions, they don't have these abrupt changes, infinite limits, or any asymptotes. Thus, nonremovable discontinuities are entirely absent in polynomial equations.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers in one or more variables multiplied by coefficients. The characteristic feature of these functions is their continuous nature, making them one of the simplest forms to analyze in calculus.

Characteristics of Polynomial Functions include:
- They are continuous for all real numbers, from negative to positive infinity.
- They don't have gaps, holes, or asymptotes.
- The graph of a polynomial function is smooth and unbroken.

The polynomial function given, \( f(x) = x^2 - 4x + 4 \), is a quadratic polynomial, meaning it has a degree of 2. Because all polynomials, including quadratics, are continuous everywhere, there are no points where the function is not defined. This continuity property means classification of discontinuity types is unnecessary, as none exist.

Continuity of Functions

Continuity in calculus refers to a function being smooth and unbroken over its domain. A function is deemed continuous at a point \( x = a \) if the following three conditions are satisfied:
1. The function \( f(x) \) is defined at \( x = a \).
2. The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
3. The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

If a function meets these criteria over its entire domain, then it is said to be continuous everywhere. In the case of polynomial functions like \( f(x) = x^2 - 4x + 4 \), the function is continuous for all real values due to its polynomial nature. Since polynomials are composed of terms that are inherently continuous, they don't have holes, jumps, or asymptotes. Therefore, its graph is always a smooth, unbroken curve.