Problem 72 - Brief Applied Calculus

Question

Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ f(x)=\left\\{\begin{array}{ll} 5-x & \text { if } x<4 \\ 2 x-5 & \text { if } x \geq 4 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is discontinuous at \(x = 4\).

Step 1: Understand Continuity

A function is continuous at a point if it does not have any abrupt changes, holes, or jumps at that point. More precisely, the function must be defined at that point, the limit as it approaches the point must exist, and both the value of the function and the limit must be equal. We will analyze these conditions for the given function.

Step 2: Evaluate Continuity at x < 4

The function is given by \(f(x) = 5-x\) for \(x < 4\). This is a linear function and continuous in its domain since it does not have any restrictions or discontinuities for \(x < 4\).

Step 3: Evaluate Continuity at x = 4

Since the function definition changes at \(x = 4\), we need to check this point carefully. Calculate the left-hand limit as \(x\) approaches 4 from the left:\[ \lim_{{x\to 4^-}} f(x) = \lim_{{x\to 4^-}} (5-x) = 5-4 = 1 \]Calculate the right-hand limit as \(x\) approaches 4 from the right:\[ \lim_{{x\to 4^+}} f(x) = \lim_{{x\to 4^+}} (2x-5) = 2(4) - 5 = 8 - 5 = 3 \]The value of the function at \(x = 4\) using the defined rule for \(x \geq 4\) is:\[ f(4) = 2(4) - 5 = 3 \]Therefore, the left-hand limit (1) and the right-hand limit (3) are not equal, which implies a jump discontinuity.

Step 4: Evaluate Continuity at x > 4

The function is given by \(f(x) = 2x - 5\) for \(x \geq 4\), which is a linear function. This function is continuous for all \(x > 4\) since it does not have any restrictions or discontinuities.

Step 5: Conclusion: Determine Discontinuity

Since at \(x = 4\) the left-hand limit \(1\) does not equal the right-hand limit \(3\), the function is discontinuous at \(x = 4\). For \(x < 4\) and \(x > 4\), the function is continuous.

Key Concepts

Piecewise Functions

In calculus, piecewise functions are essential because they allow us to define different behaviors over different intervals of the domain.
Piecewise functions are defined by multiple sub-functions, each associated with a segment of the domain. This means that the function can "change its rule" at certain points.
For example, the function mentioned in the exercise is defined as:

• \( f(x) = 5 - x \) for \( x < 4 \)
• \( f(x) = 2x - 5 \) for \( x \geq 4 \)

This structure is useful in modeling situations where there are abrupt changes, like taxable income, shipping rates, or any problem where the output changes based on input ranges.
It's important to consider transitions between these segments, which often leads to discussions about continuity at the points where the function changes from one piece to another.

Discontinuity Points

Discontinuity points in a function occur when there is an abrupt change or gap in its graph.
Such points are critical because they indicate where a function's usual behavior "breaks."
There are several types of discontinuities:

• **Jump Discontinuity:** This occurs when the left and right hand limits at a point do not match.
• **Infinite Discontinuity:** This occurs when a function approaches infinity at a discontinuity point.
• **Removable Discontinuity:** This occurs when a hole in the graph can be "filled" by adding or redefining the function value.

In the example provided, the point of interest at \( x = 4 \) has a jump discontinuity.
As per the solution, the function differs when approaching from either side of this point, causing a "jump" in the value of the function.
This is identified by the mismatch between the left-hand limit (1) and the right-hand limit of the function (3), signaling a discontinuity at this point.

Limit Calculation

Limit calculation is a core concept in calculus used to understand the behavior of functions at given points or as they approach certain values.
Calculating limits helps us analyze the continuity and differentiability of functions, especially piecewise ones.
To calculate a limit, you observe the value that the function approaches as the input gets closer to a certain point — from both left and right sides:

• **Left-Hand Limit:** Defined as \( \lim_{{x \to a^-}} f(x) \), the value as \( x \) approaches \( a \) from the left.
• **Right-Hand Limit:** Defined as \( \lim_{{x \to a^+}} f(x) \), the value as \( x \) approaches \( a \) from the right.

In the exercise, we use limit calculations at \( x = 4 \) to identify discontinuities.
For \( x = 4 \), the two limits we calculate are:

• The left-hand limit: \( \lim_{{x\to 4^-}} (5-x) = 1 \).
• The right-hand limit and function value: \( \lim_{{x\to 4^+}} (2x-5) = 3 \) and \( f(4) = 3 \).

Because the left and right limits don’t equal each other, a jump discontinuity exists.
Understanding and calculating limits helps in visualizing and resolving such discontinuities and understanding the overall behavior of piecewise functions.