Problem 35 - Calculus

Question

In Problems \(34-37\), is the function continuous for all \(x ?\) If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=\left\\{\begin{array}{ll} |x| / x & x \neq 0 \\ 0 & x=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The function is not continuous at \( x = 0 \).

Step 1: Understand the Function Definition

The function \( f(x) \) is defined as a piecewise function: it is \( \frac{|x|}{x} \) for \( x eq 0 \) and \( 0 \) for \( x = 0 \). The expression \( \frac{|x|}{x} \) is generally used to determine the sign of \( x \), resulting in 1 for positive \( x \) and -1 for negative \( x \).

Step 2: Analyze Continuity at Non-zero Points

For any point \( x eq 0 \), the function \( \frac{|x|}{x} \) is defined and continuous because it simplifies to 1 for \( x > 0 \) and -1 for \( x < 0 \). As there is no discontinuity for \( x eq 0 \), the function is continuous at these points.

Step 3: Check Continuity at Zero

To check continuity at \( x = 0 \), we calculate the left-hand limit \( \lim_{x \to 0^-} f(x) \), the right-hand limit \( \lim_{x \to 0^+} f(x) \), and the function value \( f(0) \). For \( x < 0 \), \( f(x) = -1 \), so the left-hand limit is -1. For \( x > 0 \), \( f(x) = 1 \), so the right-hand limit is 1. The function value at zero is 0.

Step 4: Apply Definition of Continuity

A function is continuous at \( x = 0 \) if the left-hand limit, right-hand limit, and the function value at that point are all equal. Here, the left-hand limit is -1, the right-hand limit is 1, and \( f(0) = 0 \). As these values are not equal, \( f(x) \) is not continuous at \( x = 0 \).

Step 5: Conclude Discontinuity Points

Since all conditions of continuity are satisfied for \( x eq 0 \) and they fail at \( x = 0 \), the function is not continuous at \( x = 0 \) due to differing limits and a differing function value at that point.

Key Concepts

Piecewise Functions

When dealing with mathematics and specifically functions, sometimes it's helpful to break a function into pieces for simpler analysis and better understanding. A piecewise function, like the one shown in this problem, is defined by different expressions depending on the value of the independent variable, usually denoted as \( x \).

In this problem:

• The function is given by two separate expressions depending on whether \( x \) is zero or not.
• It uses different rules for \( x eq 0 \) and \( x = 0 \), making the function easier to analyze for individual segments.

Breaking down functions in this way helps in understanding how the function behaves across different values and is an essential tool in calculus and beyond.

Limits

The concept of limits is fundamental in calculus, providing the bridge to understanding continuity, especially at points where the function may be tricky or undefined. A limit examines the behavior of a function as it approaches a particular point.

For this exercise:

• To check the behavior around \( x = 0 \), we check limits from both directions.
• The left-hand limit \( \lim_{x \to 0^-} f(x) \) evaluates the value as \( x \) gets closer to zero from the negative side.
• The right-hand limit \( \lim_{x \to 0^+} f(x) \) evaluates the value as \( x \) gets closer to zero from the positive side.

These left and right limits help determine if the function approaches the same value from both directions at \( x = 0 \), which is crucial for determining continuity.

Discontinuity

Discontinuity in functions refers to points where a function is not continuous, meaning the function doesn't "flow" smoothly at certain values of \( x \). Discontinuity can arise from various reasons, such as jumps, breaks, or asymptotes.

For the function in this exercise, \( f(x) \) exhibits discontinuity at \( x = 0 \):

• The left-hand limit at \( x = 0 \) is \(-1\), the right-hand limit is \(1\), and the function value is \(0\).
• Because these values don't match up, there's a distinctive jump or break at \( x = 0 \).

This illustrates a common type of discontinuity known as a jump discontinuity, where different approaches to \( x = 0 \) result in different values, causing a break in the graph.

Definition of Continuity

Understanding continuity involves examining if a function remains unbroken or consistent at a point of interest. A function is considered continuous at a particular point if it satisfies these three conditions:

• The function is defined at the point.
• The left-hand limit at the point equals the right-hand limit.
• The limit equals the actual function value at that point.

In our problem with \( f(x) \), these conditions are put to the test at \( x = 0 \):

• The function is indeed defined since \( f(0) = 0 \).
• However, \( \lim_{x \to 0^-} f(x) = -1 \) and \( \lim_{x \to 0^+} f(x) = 1 \), illustrating that the limits aren't equal, hence failing the continuity test.
• The actual function value isn't equal to these limits either.

This means that \( f(x) \) is not continuous at \( x = 0 \), which profoundly highlights how important these conditions are when scrutinizing a function's behavior.