Problem 71 - Brief Applied Calculus

Question

Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} 3-x & \text { if } x \leq 4 \\ 10-2 x & \text { if } x>4 \end{array}\right.\\\ \text { [Hint: See Exercise 37.] } \end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the pieces of the function

The function is given as a piecewise function: \( f(x) = 3 - x \) if \( x \leq 4 \) and \( f(x) = 10 - 2x \) if \( x > 4 \). We will check for continuity at the point where the pieces meet, which is \( x = 4 \).

Step 2: Evaluate the left-hand limit as x approaches 4

Calculate the left-hand limit where \( x \leq 4 \), so we use \( f(x) = 3-x \). As \( x \to 4^- \), \( \lim_{x \to 4^-} f(x) = 3 - 4 = -1 \).

Step 3: Evaluate the right-hand limit as x approaches 4

Calculate the right-hand limit where \( x > 4 \), so we use \( f(x) = 10 - 2x \). As \( x \to 4^+ \), \( \lim_{x \to 4^+} f(x) = 10 - 2(4) = 2 \).

Step 4: Evaluate f(4)

Since \( x = 4 \) is part of the domain where \( x \leq 4 \), we use \( f(x) = 3 - x \). Thus, \( f(4) = 3 - 4 = -1 \).

Step 5: Compare limits and function value at x=4

For continuity at \( x = 4 \), the left-hand limit, right-hand limit, and \( f(4) \) must all be equal. Here, \( \lim_{x \to 4^-} f(x) = -1 \), \( \lim_{x \to 4^+} f(x) = 2 \), and \( f(4) = -1 \). Since the limits are not equal, the function is discontinuous at \( x = 4 \).

Step 6: State the result of discontinuity

The function is discontinuous at \( x = 4 \) because the left-hand limit, \( f(4) \), and the right-hand limit are not all equal.

Key Concepts

Piecewise Functions

Piecewise functions are special kinds of functions that use different expressions or rules for different intervals of their domain. These functions essentially "piece" together multiple sub-functions to cover the entire domain.

Consider the function given in the exercise:

• For values where \( x \leq 4 \), the function is expressed as \( 3 - x \), meaning any input below or equal to 4 uses this formula to determine the output.
• For values where \( x > 4 \), the function switches to a different formula: \( 10 - 2x \).

Understanding piecewise functions is essential for determining how they behave at the points where the pieces change or meet. In many cases, the task is to analyze the transition at these boundary points. By doing so, we can determine if the function is continuous at these junctions or if there are any breaks or jumps in its graph.

Limits and Continuity

Limits and continuity are closely related concepts in calculus that help us understand the behavior of functions at specific points or as they approach specific values.

To determine continuity at a point, the following must hold:

• The left-hand limit, as the function approaches the point from the left, must exist.
• The right-hand limit, as the function approaches from the right, must also exist.
• The actual value of the function at that point should equal both the left-hand and right-hand limits.

In simpler words, if these three values are the same, the function is continuous at that point.

In the exercise, evaluating the limits around \( x = 4 \) showed that the left-hand limit \( \lim_{x \to 4^-} f(x) = -1 \) and the right-hand limit \( \lim_{x \to 4^+} f(x) = 2 \) are not equal. This discrepancy causes the piecewise function to be discontinuous at \( x = 4 \).

Discontinuous Functions

Discontinuous functions are ones that have breaks, jumps, or holes at certain points in their domain. These interruptions in the graph of the function mean that they aren't smooth or unbroken throughout the entire domain.

A discontinuity occurs when the criteria for continuity are not met. Such as when the left-hand and right-hand limits do not match, or if they're not equal to the actual function’s value at that point.

In the step-by-step solution of the exercise, the piecewise function:

• It was evaluated that \( x = 4 \) is a point of discontinuity because the left-hand limit and the right-hand limit are different.
• This type of discontinuity is specifically a "jump discontinuity," where the function has two different values that it approaches from either side of the point.

Understanding discontinuities is crucial, especially when graphing functions or solving real-world problems, as they indicate where a function changes behavior, or fails to follow a 'normal' expected pattern.