Question

Prove that if \(f(x)\) is a continuous function on an interval then so is the function \(|f(x)|=\sqrt{(f(x))^{2}}\).

Short answer

If \(f(x)\) is continuous, then \(|f(x)|\) is continuous because it is a composition of continuous functions.

Step-by-step solution

Understanding Continuous Functions

A function \(f(x)\) is said to be continuous on an interval if, roughly speaking, its graph can be drawn without lifting the pencil from the paper. Mathematically, \(f\) is continuous at a point \(x = c\) if \(\lim_{{x \to c}} f(x) = f(c)\). The question asks if \(|f(x)|\) is also continuous if \(f(x)\) is continuous.

Express the Absolute Value Function

The absolute value of \(f(x)\), written as \(|f(x)|\), can be expressed as \(\sqrt{{(f(x))^2}}\). This representation is helpful because it links the original function \(f(x)\) to its absolute value form, emphasizing that \(|f(x)|\) involves operations (squaring and square-rooting) on \(f(x)\).

Explore Continuity of Function Modifications

The squaring function, \((f(x))^2\), is continuous if \(f(x)\) is continuous because for any \(x\), the limit \(\lim_{{x \to c}} (f(x))^2 = (\lim_{{x \to c}} f(x))^2\) due to the continuity and limit laws. Similarly, the square root function \(\sqrt{x}\) is continuous for \(x \geq 0\). Thus, \(\sqrt{{(f(x))^2}}\) is continuous since it is the composition of continuous functions, \((f(x))^2\) and \(\sqrt{x}\).

Apply the Composition of Functions

Since both the squaring function and the square root function are continuous where they are defined (which is all of \(\mathbb{R}\) for squares and all non-negative numbers for the square root), the composition \(\sqrt{{(f(x))^2}} = |f(x)|\) is continuous wherever \(f(x)\) is continuous. This uses the fact that continuous functions composed with other continuous functions remain continuous.

Key concepts

Composition of Functions

When dealing with the continuity of functions, understanding the composition of functions is essential. A composition of functions involves creating a new function by applying one function to the results of another. For example, if you have two functions, say \(g(x)\) and \(h(x)\), the composition is denoted by \((g \circ h)(x)\), which means you first apply \(h(x)\) before \(g(x)\).
In our exercise, the absolute value function \(|f(x)|\) is expressed as \(\sqrt{(f(x))^2}\). This is a composition of the square function, \((f(x))^2\), and the square root function, \(\sqrt{x}\).

• The square function modifies \(f(x)\) into \((f(x))^2\).
• Then the square root function takes \((f(x))^2\) and returns \(\sqrt{(f(x))^2}\).

Since both these functions are continuous in their domains, their composition is continuous. This approach assures us that the operation on \(f(x)\) maintains the original function's continuity unless otherwise restricted by domain issues.
Continuity in composition is guaranteed, meaning if both functions in the composite are continuous, so is the resulting function.

Absolute Value Function

The absolute value function, often denoted as \(|f(x)|\), is a crucial element of this exercise. Understanding what the absolute value function does helps in proving its continuity when the initial function \(f(x)\) is continuous.
The absolute value of a number is its distance from zero on the number line. For any real number \(x\), the absolute value is represented as:
\[|x| = \begin{cases} x, & \text{if } x \geq 0 \-x, & \text{if } x < 0 \end{cases} \]
Here, when applying this to a function, \(|f(x)|\) transforms \(f(x)\) to non-negative values throughout its range. In this exercise, we use the expression \(\sqrt{{(f(x))^2}}\), which inherently returns a non-negative value as it squares \(f(x)\) and then applies a square root.
The transformation ensures you're modifying \(f(x)\) without impacting its continuity, as the operations of squaring and taking the square root are defined and continuous over their typical domains.

Continuous Function Properties

Continuity is a fundamental property in calculus describing functions whose graphs can be traced without lifting a pencil, representing no jumps, gaps, or asymptotes.
A function \(f(x)\) is continuous at a point \(x = c\) if three conditions are fulfilled:

• \(f(c)\) is defined.
• The limit \(\lim_{{x \to c}} f(x)\) exists.
• The limit equals the function value at that point: \(\lim_{{x \to c}} f(x) = f(c)\).

Translating this to our problem, if \(f(x)\) is continuous over an interval, every small change in \(x\) should produce a small change in \(f(x)\), without any sudden jumps.
In the context of problems involving the manipulation of \(f(x)\) via the absolute value transformation \(|f(x)|\), understanding these properties is crucial. We have already seen that the operations used to obtain the absolute value, namely squaring and taking the square root, preserve such small incremental changes, ensuring no breaks in continuity.
This inherent nature of maintaining continuity under composition showcases why such transformations in functions align well with intuitive and mathematical properties.