Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\left\\{\begin{array}{ll}\frac{1}{2} x+1, & x \leq 2 \\ 3-x, & x>2\end{array}\right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is continuous on the intervals \(-\infty < x < 2\) and \(2 < x < \infty\). It is discontinuous at the point \(x = 2\) due to a jump discontinuity. The conditions for continuity are not satisfied at \(x = 2\) as the left-hand limit and the right-hand limit of the function at \(x = 2\) are not equal.

Step 1: Understand the concept of continuity

Continuity at a point means that the limit of the function at that point exists, the function is defined at the point and the limit of the function at that point is equal to the value of the function at the point. Mathematically, if \(a\) is the point, then the function \(f(x)\) is said to be continuous at \(x=a\) if \(\lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x) = f(a)\). In case all points in an interval satisfy these conditions, the function is continuous over that interval.

Step 2: Check for continuity in the first part of the function

The first part of the function \(f(x) = \frac{1}{2}x + 1\) is defined for \(x \leq 2\). This function is a simple linear equation which is known to be continuous everywhere in its domain. Hence, this function is continuous in the interval \(-\infty < x \leq 2\).

Step 3: Checking for continuity in the second part of the function

The second part of the function \(f(x) = 3 - x\) is defined for \(x > 2\). Again, this is a linear equation and is continuous everywhere in its domain. Hence, this part of the function is continuous in the interval \(2 < x < \infty\).

Step 4: Checking for continuity at \(x = 2\)

We check for continuity at the point \(x = 2\) where the two parts of the function meet. We compute the value of the function at \(x = 2\) for both parts of the function. \(f(2) = \frac{1}{2}*2 + 1 = 2\) for the first part, and \(f(2) = 3 - 2 = 1\) for the second part. These values are different, which means this function is discontinuous at \(x = 2\), causing a jump discontinuity.

Step 5: Identify the conditions not met

The condition for continuity that is not satisfied at \(x = 2\) is that the limit from the left side is not equal to the limit from the right side. In other words, \(\lim_{{x \to 2^-}} f(x) \neq \lim_{{x \to 2^+}} f(x)\). So, continuity fails at \(x = 2\).

Key Concepts

Intervals of Continuity

Understanding the intervals of continuity involves checking where a function remains uninterrupted or smooth across its domain. In the case of our piecewise function \(f(x)\):

• The first part, \(f(x) = \frac{1}{2}x + 1\), is continuous as it is a linear function, with no breaks or gaps for all \(x \leq 2\).
• The second part, \(f(x) = 3 - x\), is also continuous for \(x > 2\), because it is linear and maintains continuity across its domain.

There is, however, a point of interest at \(x = 2\), which serves as a boundary between these intervals. Here, the question revolves around what happens when \(x\) approaches 2 from both sides. It's crucial to investigate this point thoroughly to understand why discontinuity appears and how it affects the function. Continuous intervals are important as they indicate where there's a smooth behavior of the function values.

Piecewise Functions

Piecewise functions are functions that have different expressions based on the input value \(x\). They are made up of "pieces," each defined by a specific condition. For our function, \(f(x)\), we have two parts:

• \(f(x) = \frac{1}{2}x + 1\) when \(x \leq 2\)
• \(f(x) = 3 - x\) when \(x > 2\)

Such functions can seem tricky at points where their different parts meet, as direct substitution may not be sufficient to ensure smooth transition. One vital aspect when dealing with piecewise functions is to evaluate each section individually, including at the boundaries where the function components meet, like our \(x = 2\). This analysis helps determine whether the function remains continuous or if any jumps or breaks occur at those boundary points.

Conditions of Continuity

The most basic condition for a function to be continuous at any given point \(a\) is when:

• The function \(f(x)\) is defined at \(x = a\).
• The limit of \(f(x)\) as \(x\) approaches \(a\) from the left (\(x \to a^-\)) equals the limit as \(x\) approaches from the right (\(x \to a^+\)).
• The value from both sides is equal to \(f(a)\).

In situations where any of these conditions fail to meet, as seen at \(x = 2\) in our example, we find a discontinuity, specifically called a jump discontinuity. With \(f(x)\), the limits from the left, \(\lim_{{x \to 2^-}} f(x)\), and from the right, \(\lim_{{x \to 2^+}} f(x)\), do not match. This discrepancy leads to the discontinuity at \(x = 2\), a point that disrupts the flow of the function’s established pattern in the intervals on either side.