Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 39-44, determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} 2 x-4 & \text { if } x \leq 0 \\ 1 & \text { if } x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is discontinuous at \(x = 0\). At this point, the condition \(\lim_{x \to 0^{-}} f(x) = \lim_{x \to 0^{+}} f(x)\) is violated, as the left and right limits are not equal (-4 ≠ 1).

Step 1: 1. Analyze the branch for x ≤ 0

For the function: \(f(x) = 2x - 4\) if \(x \leq 0\), we will find the limit as x approaches 0 from the left side.

Step 2: 2. Calculate the limit from the left

We want to find the limit as x approaches 0 from the left: \[\lim_{x \to 0^{-}} (2x - 4)\] Plugging in 0, we get: \[\lim_{x \to 0^{-}} (2(0) - 4) = -4\]

Step 3: 3. Analyze the branch for x > 0

For the function: \(f(x) = 1\) if \(x > 0\), we will find the limit as x approaches 0 from the right side.

Step 4: 4. Calculate the limit from the right

We want to find the limit as x approaches 0 from the right: \[\lim_{x \to 0^{+}} (1) = 1\]

Step 5: 5. Compare the left and right limits

Considering the limit from both sides of x = 0: \[\lim_{x \to 0^{-}} (f(x)) = -4\] \[\lim_{x \to 0^{+}} (f(x)) = 1\] Since the limit of the function as x approaches 0 from the left is not equal to the limit from the right, the function is discontinuous at x = 0.

Step 6: 6. State the continuity conditions violated

At x = 0, the function f(x) violates the condition: \(\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x)\), as the left and right limits are not equal.

Key Concepts

Limits

When dealing with piecewise functions, understanding limits is essential for determining points of discontinuity. A limit refers to the value that a function approaches as the input approaches some value. In the context of our piecewise function, we examine the behavior as the input, \(x\), nears 0 from both sides.
In this function, the left-hand limit as \(x\) approaches 0 from the left is calculated by plugging into \(2x - 4\). This is expressed as \(\lim_{{x \to 0^-}} (2x - 4) = -4\).
On the right side, because the function is a constant 1 for \(x > 0\), the right-hand limit as \(x\) approaches 0 is \(\lim_{{x \to 0^+}} (1) = 1\).
The lack of agreement between these two limits indicates a jump discontinuity at \(x = 0\). Limits help us understand this behavior by confirming that the function does not conform to the expected continuity at a particular point.

Continuity Conditions

To grasp why a function is discontinuous at certain points, we rely on the continuity conditions. A function \(f(x)\) is continuous at a point \(x = a\) if:

• \(f(a)\) is defined,
• the limit \(\lim_{{x \to a}} f(x)\) exists, and
• the limit matches the function value, \(\lim_{{x \to a}} f(x) = f(a)\).

In our piecewise function, evaluating the discontinuity at \(x = 0\) means comparing the limits from both sides. Since we found \(\lim_{{x \to 0^-}} f(x) = -4\) and \(\lim_{{x \to 0^+}} f(x) = 1\), the essential condition \(\lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x)\) is violated. Hence, it leads to a discontinuity. This type of continuity check is crucial to identify points where a function changes abruptly or doesn't smoothly connect.

Piecewise Functions

Piecewise functions are a type of function composed of multiple sub-functions, each defined on a certain interval of the domain. They are particularly useful for modeling scenarios where a rule or formula changes its behavior at certain points. For example, the exercise function is defined differently for \(x \leq 0\) and \(x > 0\).
Understanding piecewise functions requires paying attention to each segment and how they transition at the boundaries. Here, the segments are:\

• \(f(x) = 2x - 4\) for \(x \leq 0\), and
• \(f(x) = 1\) for \(x > 0\).

At the point \(x = 0\), the behavior of the function shifts from a linear equation to a constant value, causing a discontinuity if the transition isn't smooth. Analyzing how these segments meet highlights whether the overall function maintains continuity or if there are abrupt changes.