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Chapter 2: Problem 11

Determine the points at which the following functions \(f\) have discontinuities. For each point state the conditions in the continuity checklist that are violated.

Short Answer

Expert verified
Question: Identify the points at which a given function has discontinuities and the conditions in the continuity checklist that are violated. Answer: To determine the points at which a given function has discontinuities and the conditions in the continuity checklist that are violated, analyze and simplify the function if possible. Once potential discontinuities are identified, check against the continuity checklist which consists of 1) The function being defined at the given point, 2) The function's limit existing at the given point, and 3) The function's value at the given point being equal to its limit at that point.

Step by step solution

01

Identify given function(s)

In the exercise, we've been given a function, \(f(x)\). Unfortunately, the specific function is not provided. Without the function, it is not possible to determine the points at which it has discontinuities or the conditions in the continuity checklist that are violated. However, to help you understand how to approach such an exercise, we can consider some examples. Example 1: $$f(x) = \frac{x^2 - 1}{x - 1}$$ Example 2: $$f(x) = \frac{1}{x}$$
02

Analyze function(s), and determine discontinuities (Example 1)

We first simplify the function \(f(x) = \frac{x^2 - 1}{x - 1}\) by factoring the numerator: $$f(x) = \frac{(x - 1)(x + 1)}{x - 1}$$ Now, we can cancel out the common factors: $$f(x) = x + 1$$ But, we must remember that the original function has a restriction, \(x \neq 1\). Therefore, the function is continuous for all \(x \in \mathbb{R}\) except \(x = 1\). At \(x = 1\), the function has a discontinuity. Conditions in the continuity checklist violated for Example 1: 1. The function is not defined at \(x = 1\) (since \(x - 1\) is in the denominator).
03

Analyze function(s), and determine discontinuities (Example 2)

Now, let's analyze the second function, \(f(x) = \frac{1}{x}\). We see that this function is only undefined when \(x = 0\). At all other points, it satisfies the continuity checklist. Therefore, it has a discontinuity at \(x = 0\). Conditions in the continuity checklist violated for Example 2: 1. The function is not defined at \(x = 0\) (since \(x\) is in the denominator). In conclusion, to determine the points at which a given function has discontinuities and the conditions in the continuity checklist that are violated, the function must be analyzed and simplified if possible. Once potential discontinuities are identified, check against the continuity checklist to see which conditions are violated.

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