Problem 12 - Calculus with Concepts in Calculus

Question

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\left\\{\begin{array}{rl} -1 & \text { for } x<0 \\ 0 & \text { for } x=0 \\ 1 & \text { for } x>0 \end{array} \quad a=0\right. $$

Step-by-Step Solution

Answer

The function \( f \) is discontinuous at \( a = 0 \); it is neither continuous from the left nor the right at \( a = 0 \).

Step 1: Identify the function and the point of interest

We have a piecewise function: \( f(x) = \begin{cases} -1 & \text{for } x < 0 \ 0 & \text{for } x = 0 \ 1 & \text{for } x > 0 \end{cases} \) and we need to check its continuity at the point \( a = 0 \).

Step 2: Determine left-hand limit \( \lim_{{x \to 0^-}} f(x) \)

To find the left-hand limit as \( x \to 0^- \), evaluate \( f(x) \) for values slightly less than 0. The expression for \( x < 0 \) is \( f(x) = -1 \). Therefore, \( \lim_{{x \to 0^-}} f(x) = -1 \).

Step 3: Determine right-hand limit \( \lim_{{x \to 0^+}} f(x) \)

To find the right-hand limit as \( x \to 0^+ \), evaluate \( f(x) \) for values slightly greater than 0. The expression for \( x > 0 \) is \( f(x) = 1 \). Therefore, \( \lim_{{x \to 0^+}} f(x) = 1 \).

Step 4: Evaluate the function at \( a = 0 \)

The value of the function at \( a = 0 \) is given directly by the piecewise function, \( f(0) = 0 \).

Step 5: Determine overall limit \( \lim_{{x \to 0}} f(x) \)

For \( f(x) \) to be continuous at \( a = 0 \), \( \lim_{{x \to 0^-}} f(x) = \lim_{{x \to 0^+}} f(x) = f(0) \). However, \( \lim_{{x \to 0^-}} f(x) = -1 \) and \( \lim_{{x \to 0^+}} f(x) = 1 \). These limits are not equal, so \( \lim_{{x \to 0}} f(x) \) does not exist.

Step 6: Determine continuity types at \( a = 0 \)

Since \( \lim_{{x \to 0}} f(x) \) does not exist, \( f(x) \) is discontinuous at \( a = 0 \). The left-hand limit \( \lim_{{x \to 0^-}} f(x) eq f(0) \), making it discontinuous from the left. The right-hand limit \( \lim_{{x \to 0^+}} f(x) eq f(0) \), making it discontinuous from the right as well.

Key Concepts

Understanding Piecewise Functions

A piecewise function is a function defined by different expressions for different intervals of its domain. This means that instead of having one continuous rule applying to all values, the function has different rules depending on the value of the input. For example, in the given exercise, \( f(x) \) is defined differently for values that are less than 0, equal to 0, and greater than 0:

• For \( x < 0 \), \( f(x) = -1 \)
• For \( x = 0 \), \( f(x) = 0 \)
• For \( x > 0 \), \( f(x) = 1 \)

Piecewise functions are useful for expressing mathematical models of situations with changes at specific points, like tax brackets or shipping rates. In analyzing piecewise functions, especially at points where the rule changes, you often need to consider limits.

The Concept of Limits

Limits help us understand the behavior of functions as inputs approach a specific value. In calculus, they are used to find where a function might "tend towards" as inputs get infinitesimally close to a certain number.To determine if a function is continuous at a certain point, we typically check:

• The limit from the left (left-hand limit).
• The limit from the right (right-hand limit).
• The actual value of the function at that point.

For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function's value at that point must all be equal. If they differ, the function is said to be discontinuous at that point. In the exercise, we examined \( f(x) \) as \( x \) approaches 0. We found that the limit from the left is different from the limit from the right, indicating discontinuity.

Exploring Left-Hand Limits

When determining a left-hand limit, we focus on the value that \( f(x) \) approaches as \( x \) gets closer to a specific point from the left side exclusively. Represented as \( \lim_{{x \to a^-}} f(x) \), it checks for values slightly less than \( a \). In the given example, as \( x \) approaches 0 from the left, \( f(x) = -1 \). Hence, the left-hand limit is \( \lim_{{x \to 0^-}} f(x) = -1 \). This limit helps us anticipate what the function's outputs are near a boundary from a particular direction, and together with right-hand limits, decides the continuity at that point.

Examining Right-Hand Limits

The right-hand limit looks at what \( f(x) \) approaches as the variable \( x \) nears a specific point from the right side. It is denoted as \( \lim_{{x \to a^+}} f(x) \) and involves assessing the function for values slightly greater than \( a \).In the example exercise, as we approach 0 from the right, \( f(x) = 1 \). Thus, the right-hand limit here is \( \lim_{{x \to 0^+}} f(x) = 1 \). The observation of right-hand limits, alongside left-hand limits, is essential for understanding whether a function's behavior smoothly transitions through a point or if there are abrupt changes, as seen in this discontinuity.