Question

Prove that the absolute value function \(|x|\) is continuous for all values of \(x\). (Hint: Using the definition of the absolute value function, compute \(\left.\lim _{x \rightarrow 0^{-}}|x| \text { and } \lim _{x \rightarrow 0^{+}}|x| .\right)\)

Short answer

Answer: Yes, the absolute value function is continuous for all values of x.

Step-by-step solution

Definition of continuity

For a function to be continuous at a point, the following conditions must be satisfied: 1. The value of the function must be defined at the point. 2. The left-hand limit must be equal to the right-hand limit. 3. The limit must be equal to the value of the function at the point. We will now examine each condition for the absolute value function, \(|x|\).

Value of the function at x=0

The absolute value function is defined as: $$ |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases} $$ At \(x = 0\), the function is equal to \(|0| = 0\). This satisfies the first condition.

Compute left-hand limit

Let's compute the left-hand limit of the function as \(x\) approaches \(0\) from the left side: $$ \lim_{x \rightarrow 0^{-}} |x| = \lim_{x \rightarrow 0^{-}}(-x) $$ Since \(x\) is negative for values approaching \(0\) from the left, we can substitute \(h = -x\) where \(h \rightarrow 0\); and now: $$ \lim_{h \rightarrow 0} h = 0 $$ So, the left-hand limit at \(x = 0\) is 0.

Compute right-hand limit

Now, let's compute the right-hand limit of the function as \(x\) approaches \(0\) from the right side: $$ \lim_{x \rightarrow 0^{+}} |x| = \lim_{x \rightarrow 0^{+}}(x) $$ Since \(x\) is positive for values approaching \(0\) from the right, we can again substitute \(h = x\) where \(h \rightarrow 0\): $$ \lim_{h \rightarrow 0} h = 0 $$ So, the right-hand limit at \(x = 0\) is 0.

Check continuity at x=0

Now that we have computed the left-hand limit, right-hand limit, and the value of the function at \(x = 0\), let's check if they are equal: The left-hand limit is 0, the right-hand limit is 0, and the function's value at \(x = 0\) is also 0. Since all three are equal, the absolute value function is continuous at \(x = 0\).

Conclusion

Given that the absolute value function is symmetric and behaves the same on both sides of \(x = 0\), proving its continuity at \(x = 0\) guarantees its continuity for all other values of \(x\) as well. Thus, we have proved that the absolute value function \(|x|\) is continuous for all values of \(x\).

Key concepts

Step-by-step solution

Breaking down a problem into smaller, manageable steps is often the best way to understand complex concepts. In proving the continuity of the absolute value function, step-by-step analysis simplifies the process.

• First, we reviewed the definition of continuity requiring the value of a function at a point, its defining limit from the left, and its limit from the right all equal to each other.
• Next, for \(x = 0\), we found that the absolute value function's value is \(|0| = 0\).
• By calculating both the left-hand and right-hand limits as \(x\) approaches 0, each equaled to 0, confirming equal limits at that point.
• Finally, these steps confirmed that \(|x|\) is continuous at \(x = 0\), ensuring its continuity overall, thanks to the function's symmetric nature.

All taken together, this breakdown affirms the process is both systematic and rigorous, which helps in grasping the concept fully.

Limits

The concept of limits is a fundamental building block in calculus. It helps us understand the behavior of functions as inputs approach a particular value. In analyzing the continuity of the absolute value function, we calculated limits from both directions as \(x\) approached 0.

• The left-hand limit, as \(x\) approaches 0 from the negative side, shows that the behavior is effectively like the negative input negating into a positive value, thus leading the limit to be 0.
• The right-hand limit similarly aligns, starting from positive inputs and smoothly transitioning into 0 without deviation.

Thus, both the left-hand and right-hand limits being equal substantiates a function's potential continuity at that specific point.

Absolute value function

The absolute value function, represented by \(|x|\), provides significant insights into mathematical analyses by representing distance on a number line.

• Mathematically, \(|x|\) is defined as \(x\) if \(x\) is non-negative (\(x \geq 0\)), and \(-x\) if \(x\) is negative (\(x < 0\)). Thus, the resulting output is non-negative regardless of the input.
• Graphically, the absolute value function forms a 'V' shape centered at the origin, showcasing symmetry on either side of \(x = 0\).
• This inherent symmetry ensures that limits calculated on either side of any point, particularly 0, are equal. Thus, these properties make the absolute value function easily verified as continuous across all real numbers.

By understanding how it converts negative inputs into non-negative outputs, and how it maintains uniform progression, we appreciate why it's classified as a continuous function across its domain.