Problem 37 - 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? \ $$ f(x)=x^{2}-9 $$

Step-by-Step Solution

Answer

There are no discontinuities in the given function. Therefore, there are no removable or nonremovable discontinuities.

Step 1: Identifying the Type of Function

The given function \(f(x) = x^2 - 9\) is a polynomial function which is a smooth continuous curve everywhere for the values of \(x\).

Step 2: Graphical Representation for Discontinuity

Draw the graph of the function \(f(x) = x^2 - 9\). The graph is a parabola which opens upward and is continuous at every point.

Step 3: Identifying the Discontinuity

On observing the graph, there is clearly no discontinuity in the graph of function. It does not have any removable or nonremovable discontinuities because they typically occur in rational functions.

Step 4: Conclusion

Therefore, the polynomial function \(f(x) = x^2 - 9\) is continuous everywhere and does not have any discontinuities.

Key Concepts

Removable Discontinuity

A removable discontinuity occurs when a function has a point at which its value is either undefined or does not match the overall behavior of the function, but it can be 'fixed' or made continuous by redefining the function at that point.

For example, the function given by the equation \( f(x) = \frac{{(x-3)(x+3)}}{{(x-3)}} \) has a removable discontinuity at \( x = 3 \) because the function is not defined there due to division by zero. However, the factor \( (x-3) \) in the numerator and denominator cancels out, and if we redefine the function as \( f(x) = x + 3 \) when \( x = 3 \), the discontinuity is 'removed'. The graph of this function would show a hole at the point \( (3, 6) \), but otherwise it would be a continuous line.

Nonremovable Discontinuity

Nonremovable discontinuities are the points at which a function is inherently discontinuous, and this type of discontinuity cannot be 'fixed' by simply redefining the function at those points.

There are several types of nonremovable discontinuities, including jump discontinuities, where the function has a sudden jump in values; infinite discontinuities, where the function goes to infinity; or oscillating discontinuities, where the function has an indefinite form. An example of a function with a nonremovable discontinuity is \( f(x) = \frac{1}{x} \), which has an infinite discontinuity at \( x = 0 \) because the function goes to infinity as x approaches zero.

Continuous Functions

Continuous functions are those functions that have no discontinuities. This means you can draw the graph of a continuous function without lifting your pencil from the paper.

For a function \( f(x) \) to be continuous at a point \( x = a \), three conditions must be met:

• The function must be defined at \( x = a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) must exist.
• The limit of \( f(x) \) as \( x \) approaches \( a \) must be equal to \( f(a) \).

If any of these conditions are not met, the function is considered discontinuous at that point. Polynomial functions like \( f(x) = x^2 - 9 \) are classic examples of continuous functions as they meet the criteria for continuity at every point in their domain.

Polynomial Functions

Polynomial functions are expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An important property of polynomial functions is that they are continuous and smooth everywhere in their domain, which consists of all real numbers.

For instance, the function \( f(x) = x^2 - 9 \) is a polynomial of degree 2 (a quadratic function) and its graph is a parabola. Since polynomials do not have variables in the denominator or any undefined operations, they do not introduce removable or nonremovable discontinuities. Thus, their graphs do not exhibit gaps, jumps, or vertical asymptotes.