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Chapter 2: Problem 174

Prove the following functions are continuous everywhere $$ g(x)=|x| $$

Short Answer

Expert verified
The function \( g(x) = |x| \) is continuous everywhere.

Step by step solution

01

Understand the Definition of Continuity

A function \( f \) is continuous at a point \( x = a \) if the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \). Formally, this means \( \lim_{x \to a} f(x) = f(a) \). For \( g(x) = |x| \) to be continuous everywhere, it must be continuous at every real number \( a \).
02

Evaluate Continuity for x > 0

For \( x > 0, \ g(x) = x \), which is a simple linear function. Linear functions are continuous for all real numbers. Therefore, \( g(x) \) is continuous for all \( x > 0 \).
03

Evaluate Continuity for x < 0

For \( x < 0, \ g(x) = -x \), which is also a linear function. Since linear functions are continuous for all real numbers, \( g(x) \) is continuous for all \( x < 0 \).
04

Evaluate Continuity at x = 0

At \( x = 0, \ g(0) = |0| = 0 \). We need to check \( \lim_{x \to 0} g(x) = g(0) \). As \( x \to 0^+ \) (approaching from the right), \( g(x) = x \to 0 \). As \( x \to 0^- \) (approaching from the left), \( g(x) = -x \to 0 \). Since both limits equal \( g(0) = 0 \), \( g(x) \) is continuous at \( x = 0 \).
05

Conclude Continuity Everywhere

Since \( g(x) = |x| \) is continuous for \( x > 0 \), \( x < 0 \), and at \( x = 0 \), it follows that \( g(x) \) is continuous everywhere on the real line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
The absolute value function, denoted as \( g(x) = |x| \), is a function that outputs the non-negative value of any number \( x \). It is defined piecewise as follows:
  • For \( x \geq 0 \), \( |x| = x \)
  • For \( x < 0 \), \( |x| = -x \)
The absolute value function is graphically represented as a "V" shape, which pivots at the origin (0,0). This function is commonly used to describe distances and magnitudes since it always provides positive outputs or zero. In calculus, the absolute value is especially important in defining the distance between points on the number line or analyzing real-life situations where direction is not significant, just magnitude.
Limit in Calculus
The limit concept in calculus is foundational for understanding continuity and differentiability. The limit describes the behavior of a function as its input approaches a particular value. We write the limit of a function \( f(x) \) as \( x \) approaches \( a \) using the notation \( \lim_{x \to a} f(x) \). For the absolute value function \( g(x)=|x| \), evaluating limits was a key part of proving continuity. By examining \( \lim_{x \to a} g(x) \), we determine if the value of the function \( g(x) \) is equal to \( g(a) \), ensuring the function has no gaps or jumps at \( a \).Understanding Limits: Key Points
  • The limit must exist and be the same when approached from both the left and the right of \( a \).
  • If \( \lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = f(a) \), then \( f(x) \) is continuous at \( x = a \).
Piecewise Linear Function
A piecewise linear function consists of two or more linear segments. Each segment defines the function over a specific interval. For \( g(x)=|x| \), where
  • For \( x \geq 0 \), \( g(x) = x \), which is a linear function with a slope of 1,
  • For \( x < 0 \), \( g(x) = -x \), which is a linear function with a slope of -1.
These distinct linear parts combine to form the absolute value function, showcasing a change in direction but maintaining continuity across the domain. The piecewise nature of \( g(x) \) results in the sharp vertex at \( x = 0 \), creating the characteristic "V" shape. Understanding piecewise functions is crucial for analyzing functions with different rules over different intervals, which is common in mathematical problems and real-world applications.
Real Analysis
Real analysis is a branch of mathematics dealing with real numbers and real-valued functions. It includes the study of sequences, series, limits, continuity, differentiability, and integrability. When we discuss the continuity of \( g(x) = |x| \), we rely on the principles of real analysis to rigorously prove that a function works as expected across the entire real number line. In real analysis:
  • A function must be continuous at every point in its domain to be deemed continuous everywhere.
  • Continuity is ensured when the limit matches the function's value at each point.
Real analysis provides us with the tools to demonstrate these properties, ensuring a deep and precise understanding of how functions behave in calculus. This makes real analysis fundamental to both pure and applied mathematics.

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Most popular questions from this chapter

In the following exercises, assume that \(\lim _{x \rightarrow 6} f(x)=4, \lim _{x \rightarrow 6} g(x)=9, \quad\) and \(\lim _{x \rightarrow 6} h(x)=6 .\) Use these three facts and the limit laws to evaluate each limit. $$\lim _{x \rightarrow 6} 2 f(x) g(x)$$

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow a} f(x)=N$$

In the following exercises, assume that \(\lim _{x \rightarrow 6} f(x)=4, \lim _{x \rightarrow 6} g(x)=9, \quad\) and \(\lim _{x \rightarrow 6} h(x)=6 .\) Use these three facts and the limit laws to evaluate each limit. $$\lim _{x \rightarrow 6} \frac{g(x)-1}{f(x)}$$

$$ h(x)=\left\\{\begin{array}{ll}{3 x^{2}-4,} & {x \leq 2} \\ {5+4 x,} & {x>2}\end{array} \text { Over the interval }\right. $$ \([0,4], \quad\) there is no value of \(x\) such that \(h(x)=10\) although \(h(0)<10\) and \(h(4)>10\) . Explain why this does not contradict the IVT.

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? $$ f(x)=\left\\{\begin{array}{ll}{x \sin (x)} & {\text { if } x \leq \pi} \\ {x \tan (x) \text { if } x>\pi}\end{array}, \text { at } x=\pi\right. $$
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