Problem 94 - Calculus: Early Transcendentals

Question

Does continuity of \(|\boldsymbol{f}|\) imply continuity of \(\boldsymbol{f}\) ? Let $$g(x)=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\\-1 & \text { if } x<0\end{array}\right.$$ a. Write a formula for \(|g(x)|\) b. Is \(g\) continuous at \(x=0 ?\) Explain. c. Is \(|g|\) continuous at \(x=0 ?\) Explain. d. For any function \(f\), if \(|f|\) is continuous at \(a\), does it necessarily follow that \(f\) is continuous at \(a\) ? Explain.

Step-by-Step Solution

Answer

Explain your answer using an example. Answer: No, the continuity of the absolute value of a function does not always imply the continuity of the function itself. One example is the function g(x) = 1 if x ≥ 0, and g(x) = -1 if x < 0. The absolute value of g(x), which is 1 for all x, is continuous at x = 0, while the function g(x) itself is discontinuous at x = 0, as its left and right limits at x = 0 are not equal.

Step 1: Part a: Formula for |g(x)|

In order to write a formula for |g(x)|, we will replace g(x) with its definition and calculate the absolute value for each case: $$|g(x)|=\left\\{\begin{array}{ll}|1| & \text { if } x \geq 0 \\\\-|-1| & \text { if } x<0\end{array}\right.$$ Which simplifies to: $$|g(x)|=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\1 & \text { if } x<0\end{array}\right.$$

Step 2: Part b: Continuity of g(x) at x=0

To check if g(x) is continuous at x=0, we need to verify if the limit as x approaches 0 from the left and from the right are the same: $$\lim_{x \to 0^-} g(x) = -1$$ $$\lim_{x \to 0^+} g(x) = 1$$ Since the left and right limits are not equal, g(x) is not continuous at x=0.

Step 3: Part c: Continuity of |g(x)| at x=0

Now, let us check the continuity of |g(x)| at x=0. We need to verify if the limit as x approaches 0 from the left and from the right are the same: $$\lim_{x \to 0^-} |g(x)| = 1$$ $$\lim_{x \to 0^+} |g(x)| = 1$$ Since the left and right limits are equal, |g(x)| is continuous at x=0.

Step 4: Part d: Continuity of function f if |f| is continuous

Let's consider any function f and its absolute value |f|. If |f| is continuous at a point a, it doesn't necessarily imply that f itself is continuous at a. As demonstrated in the earlier parts of this exercise, the absolute value could "hide" the discontinuity in f, like in the case of g(x) and |g(x)| at x=0. Thus, we can conclude that the continuity of |f| doesn't necessarily imply the continuity of the function f.

Key Concepts

Absolute Value Continuity

The concept of absolute value continuity delves into understanding how a function behaves once its values are converted to their absolute magnitudes. With absolute values, we're essentially looking at the 'distance' from zero without concern for direction.

In our exercise, we observed the function g(x), which takes different signs based on the input value of x. Once we applied the absolute value to it, |g(x)|, we noticed that the resulting function becomes constant, equal to 1 for all x. This is a clear example of absolute value modifying the behavior of a function.

Here's an important takeaway: a function being continuous after applying the absolute value does not guarantee the original function is continuous. This is because the absolute value can 'smooth over' any jumps or sign changes that might signal a discontinuity in the original function. In simpler terms, while the absolute value of a function might always land softly on its feet, the original function could still take a tumble at certain points.

Limits in Calculus

The concept of limits in calculus is fundamental when discussing continuity. Limits help us understand the behavior of functions as they approach specific points. To establish whether a function is continuous at a certain point, we look at the limits from both the left and the right of that point.

If these one-sided limits don't match, as was the case with g(x) at x=0, we have found a discontinuity. For g(x), the limits are -1 from the left and 1 from the right, which clearly aren't equal and point to a 'break' in the function.

Understanding limits is essential for identifying points of continuity and discontinuity. Being able to calculate these limits is often one of the first steps in evaluating the overall continuity of a function.

Piecewise Functions

A piecewise function, like the g(x) in our exercise, is a function that has different expressions or 'pieces' based on the input. These functions can offer unique insights into continuity because each 'piece' may have its own behavior.

When checking for continuity, especially with piecewise functions, it's crucial to examine where these pieces connect—often at the boundaries of their defined intervals. g(x) shows us a simple piecewise function with a clear division at x=0, switching from -1 to 1. Those sudden changes are typical spots to scrutinize for potential discontinuities.

Piecewise functions can act as a great exercise in conceptual understanding because you're tasked with piecing together these different 'behavioral snippets' into a cohesive understanding of the function's continuity as a whole.

Continuity at a Point

Speaking of continuity at a point, there's a precise definition that must be met for a function to be considered continuous at a specific point. A function f is continuous at a point a if the following are true: The function is defined at a, the limit as x approaches a exists, and the limit equals the function's value at a.

With the function g(x), continuity at x=0 is disrupted because the limits from the left and right do not match. On the other hand, its absolute version |g(x)| is indeed continuous at x=0 since the limits agree and the function is defined there.

These rules provide a mathematical structure to analyze potential points of discontinuity in a function's domain, allowing for a methodical approach to understand and graph the behavior of different functions.