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Chapter 3: Problem 22

Let the functions \(f\) and \(g\) be defined by $$ f(x)=\left\\{\begin{array}{ll} \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array} \quad g(x)= \begin{cases}x \sin (1 / x) & \text { if } x \neq 0 \\\ 0 & \text { if } x=0\end{cases}\right. $$ Show that \(f\) is not continuous at \(x=0\) but that \(g\) is continuous. Sketch the graphs of \(f\) and \(g\).

Short Answer

Expert verified
\(f(x)\) is not continuous at \(x=0\); \(g(x)\) is continuous at \(x=0\).

Step by step solution

01

Understanding the Problem

We are given two piecewise functions \(f\) and \(g\). We need to determine the continuity of each function at \(x=0\), and we will investigate the behavior of \(f(x)\) and \(g(x)\) as \(x\) approaches 0.
02

Define Continuity

A function \(h(x)\) is continuous at a point \(a\) if \(\lim_{{x \to a}} h(x) = h(a)\). This means the limit of the function as \(x\) approaches \(a\) should equal the value of the function at \(a\). We'll apply this definition to both functions at \(x=0\).
03

Evaluate Continuity of \(f(x)\) at \(x=0\)

For \(x eq 0\), \(f(x) = \sin(1/x)\). As \(x\) approaches 0, the term \(1/x\) grows very large, causing \(\sin(1/x)\) to oscillate between \(-1\) and \(1\) rapidly. This means the limit \(\lim_{{x \to 0}} f(x)\) does not exist due to this oscillation. Therefore, \(f(x)\) is not continuous at \(x=0\).
04

Evaluate Continuity of \(g(x)\) at \(x=0\)

For \(x eq 0\), \(g(x) = x \sin(1/x)\). As \(x\) approaches 0, although \(\sin(1/x)\) oscillates between \(-1\) and \(1\), multiplying by \(x\) makes the oscillations shrink towards 0 since \(x\) itself approaches 0. Therefore, \(\lim_{{x \to 0}} g(x) = 0\), which matches \(g(0) = 0\). Thus, \(g(x)\) is continuous at \(x=0\).
05

Sketch the Graphs

For \(f(x) = \sin(1/x)\), as \(x\) approaches 0, the graph will show rapid oscillations between \(-1\) and \(1\). For \(g(x) = x \sin(1/x)\), the oscillations dampen as \(x\) approaches 0, converging on the point (0,0), giving \(g(x)\) a 'pinched' look at the origin.

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Key Concepts

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

Piecewise Functions
A piecewise function is defined by different expressions depending on the input value of the variable. It allows flexibility in defining a function that responds differently to different ranges or specific points.
In the exercise provided, we observe piecewise definitions for functions \(f(x)\) and \(g(x)\):
  • For \(f(x)\), the function takes the value \(\sin(1/x)\) when \(x eq 0\), and it becomes 0 when \(x = 0\).
  • Similarly, \(g(x)\) is defined as \(x \sin(1/x)\) for \(x eq 0\), and it is 0 for \(x = 0\).
This piecewise fashion enables functions to adopt different behaviors based on the input, which is quite common in mathematical modeling and real-world applications.
When dealing with piecewise functions, it's essential to carefully consider their definition at key points like \(x = 0\) to understand their behavior fully.
Oscillation
Oscillation refers to a repeated variation, typically in time, of some measure about a central value. It usually describes the behavior of functions that fluctuate between values without settling.
In the context of function \(f(x) = \sin(1/x)\), as \(x\) approaches zero, \(1/x\) becomes large, causing the sine function to oscillate rapidly between -1 and 1. Such behavior makes it challenging for some functions to settle at a limit.
  • Rapid oscillation near a point can cause a function to be discontinuous at that point, as the function does not approach a single value.
In the example, \(f(x)\) demonstrates this perfectly by oscillating too rapidly as \(x\) nears zero. This prevents \(f\) from being continuous at \(x=0\). However, when \(x\) is involved in \(g(x) = x \sin(1/x)\), the \(x\) factor shrinks the oscillations to zero, resulting in a continuous function.
Limits
Limits are a fundamental concept in calculus, describing the behavior of a function as it approaches a specific point.
To determine the continuity of a function at a point, such as \(x = 0\), we examine the limit of the function as \(x\) approaches that point.
  • A function is continuous at a point if the limit of the function as \(x\) approaches the point equals the function's value at that point.
For \(f(x)\), \(\lim_{{x \to 0}} \sin(1/x)\) does not exist due to oscillation. Hence, \(f(x)\) is not continuous at \(x=0\).
In contrast, \(\lim_{{x \to 0}} g(x)\) evaluates to 0 and \(g(0)\) is also 0. This ensures \(g(x)\) is continuous at \(x=0\). The manipulation of limits provides a way to assess how functions behave and ensures understanding of crucial concepts like continuity.

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Most popular questions from this chapter

Let \(n\) be a positive integer and let \(r\) be any rational number. Show that the following functions are continuous: (i) \(f(x)=x^{n} \quad(x \in \mathbb{R})\) (ii) \(g(x)=x^{-n} \quad(x \neq 0)\) (iii) \(h(x)=\sqrt[n]{x} \quad(x \geqslant 0)\) (iv) \(j(x)=x^{r} \quad(x>0)\)

A function is called strictly increasing if whenever \(x_{1}\) and \(x_{2}\) are in its domain with \(x_{1}f\left(x_{2}\right)\).) Show that a function which is strictly increasing is one-to-one and that its inverse function is also strictly increasing. Prove also the corresponding result about a decreasing function. Give an example of a one-to-one function which is neither increasing nor decreasing.

Show that the cosine function is continuous and that \(\lim _{x \rightarrow x_{0}} \frac{\cos x-\cos x_{0}}{x-x_{0}}=-\sin x_{0}\) Interpret that result in terms of gradients.

For any \(x \in R\) let \(y=\sinh ^{-1} x\). Show that \(\mathrm{e}^{2 y}-2 x \mathrm{e}^{y}-1=0\) and deduce that $$ \sinh ^{-1} x=\log \left(x+\sqrt{\left.\left(1+x^{2}\right)\right)}\right. $$

Let \(f\) be any function with domain \(\mathbb{R}\) for which: (i) \(f(1)\) is positive (call it \(a\) say); (ii) \(f(x) \times f(y)=f(x+y)\) for each pair of rationals \(x\) and \(y\). By first dealing with the integers show that \(f(x)=a^{x}\) for each rational number \(x\). Now assume in addition that \(f\) has the property that if \(x_{1}, x_{2}, x_{3}, \ldots \rightarrow x\) then \(f\left(x_{1}\right), f\left(x_{2}\right), f\left(x_{3}\right), \ldots \rightarrow f(x)\) Deduce that \(f(x)=a^{x}\) for each number \(x\).
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