Question

Properties of Continuity Prove that if \(f\) is continuous on an interval, then so is \(|f| .\)

Short answer

The proof goes as follows: if \(f\) is continuous on an interval I, then for any point \(c\) in I, for all \(ε > 0\), there exists a \(δ > 0\) such that if \(0 < |x - c| < δ\) then \(|f(x) - f(c)| < ε\). Then consider two cases - when \(f(x) ≥ 0\) and when \(f(x) < 0\). In both cases, one can show that if \(0 < |x - c| < δ\) then \(||f(x)| - |f(c)|| < ε\). Hence, by definition of continuity, \(|f|\) is also continuous at \(c\). Since \(c\) was arbitrary, \(|f|\) is continuous on \(I\).

Step-by-step solution

Define the setup

Let's assume that \(f\) is continuous on an interval \(I\). We want to prove that \(|f|\) is also continuous on \(I\). Let's take any point \(c\) in interval \(I\).

Apply definition of continuity to \(f\)

Since \(f\) is continuous at \(c\), by the definition of continuity, for all \(ε > 0\), there exists a \(δ > 0\) such that if \(0 < |x - c| < δ\) then \(|f(x) - f(c)| < ε\).

Consider the two cases

We now consider two cases: where \(f(x) ≥ 0\) and \(f(x) ≤ 0\).

Case 1: \(f(x) ≥ 0\)

If \(f(x) ≥ 0\), then \(|f(x)| = f(x)\) and \(|f(c)| = f(c)\). Hence, \(||f(x)| - |f(c)|| = |f(x) - f(c)| < ε\) whenever \(0 < |x - c| < δ\).

Case 2: \(f(x) < 0\)

If \(f(x) < 0\), then \(|f(x)| = -f(x)\) and \(|f(c)| = -f(c)\). Therefore, \(||f(x)| - |f(c)|| = |-f(x) + f(c)| = |f(x) - f(c)| < ε\) whenever \(0 < |x - c| < δ\).

Conclude the proof

In both cases, we have shown that if \(0 < |x - c| < δ\) then \(||f(x)| - |f(c)|| < ε\). Hence, by the definition of continuity, \(|f|\) is continuous at \(c\). Since \(c\) was arbitrary, we have shown that \(|f|\) is continuous on \(I\).

Key concepts

Definition of Continuity

Understanding the definition of continuity in calculus is crucial for solving problems related to continuous functions. Continuity at a point within the domain of a function means that the limit of the function as it approaches the point from either direction is equal to the function's value at that point. In mathematical terms, a function \(f\) is continuous at a point \(c\) if for every \(ε > 0\), there exists a \(δ > 0\) such that for all \(x\) within the interval \(I\), if \(0 < |x - c| < δ\), then \(|f(x) - f(c)| < ε\).
In simpler terms, this ensures that the graph of the function has no holes, jumps, or breaks at the point \(c\). It helps us predict the behavior of functions and understand how they will act under various operations, such as taking the absolute value, which is often a point of confusion but is clarified by understanding the fundamental nature of continuity.

Properties of Continuity

The properties of continuity provide a framework for understanding how continuous functions behave and interact. One such property is the preservation of continuity under certain operations. For instance, it's a well-established property that if a function \(f\) is continuous, then so will be its absolute value function \(|f|\).
This property comes from the fact that applying the absolute value operation to a function does not create any new breaks or holes in its graph; it only reflects certain parts of the graph about the x-axis. Moreover, the arithmetic of continuous functions—such as addition, subtraction, multiplication, and division (except where the denominator is zero)—preserve continuity. These properties are fundamental in proving more complex theorems and are often used to solve calculus problems.

Continuous Functions

Continuous functions are the backbone of calculus, and their significance cannot be understated. Functions that are continuous over an interval or the entire domain are easier to analyze and work with because they behave predictably. When a function is continuous, you can expect no sudden changes in its output as you slightly change the input within its domain.

Common Continuous Functions

• Polynomials
• Exponential functions
• Sine and cosine functions
• Logarithmic functions

Each of these has a domain over which they are continuous. Understanding the types of functions that are inherently continuous helps in constructing more complex functions and in analyzing the effects of various operations on continuity.

Absolute Value Functions

Absolute value functions have a characteristic 'V' shape and are a prime example of continuous functions. Defined as \(|f(x)| = f(x)\) if \(f(x) ≥ 0\) and \(|f(x)| = -f(x)\) otherwise, these functions take all real numbers as inputs and give non-negative real numbers as outputs. The key to understanding absolute value functions in the context of continuity is recognizing that they only modify the sign of the function's value, without causing any breaks or interruptions in the graph.
However, it must be noted that the vertex of the 'V' graph (where the function changes from increasing to decreasing, or vice versa) is a point at which the function's derivative is not defined, even though the absolute value function itself is continuous at that point. This highlights the distinction between continuity and differentiability, which is another essential concept in calculus.