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Magazine Chapter 1: Problem 64
Is the function continuous for all \(x ?\) If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=\left\\{\begin{array}{ll} x / x & x \neq 0 \\ 1 & x=0 \end{array}\right.$$
Short Answer
Expert verified
The function is continuous for all \( x \).
Step by step solution
01
Understanding the Function
The function given is a piecewise function. For all \( x eq 0 \), the function is \( f(x) = x/x = 1 \). At \( x = 0 \), the function is provided as \( f(x)=1 \).
02
Definition of Continuity
A function \( f(x) \) is continuous at a point \( x = a \) if the following conditions are met: 1) \( f(a) \) is defined, 2) \( \lim_{{x \to a}} f(x) \) exists, and 3) \( \lim_{{x \to a}} f(x) = f(a) \).
03
Check Continuity for \(x \neq 0\)
For all \( x eq 0 \), \( f(x) = 1 \). This function is constant and continuous everywhere except possibly at \( x = 0 \). So, we focus on checking continuity at \( x = 0 \).
04
Check Limit at \(x = 0\)
The limit as \( x \) approaches 0 of \( f(x) \), where \( f(x) = x/x = 1 \), is \( \lim_{{x \to 0}} 1 = 1 \). This limit exists.
05
Comparing Limit and Function Value at \(x = 0\)
The function's value at \( x = 0 \) is \( f(0) = 1 \). The limit as \( x \to 0 \) is also 1. Since \( \lim_{{x \to 0}} f(x) = f(0) = 1 \), the function meets the conditions for continuity at \( x = 0 \).
06
Conclusion on Continuity
Since the function \( f(x) \) is continuous for all \( x eq 0 \) and the conditions for continuity at \( x = 0 \) are satisfied, the function is continuous for all \( x \in \mathbb{R} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Piecewise function
A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval or condition. In this exercise, the function is defined in two parts:
- For all values where \( x eq 0 \), the function is \( f(x) = x/x = 1 \).
- At \( x = 0 \), the function specifically assigns \( f(x) = 1 \).
Limit of a function
The limit of a function as \( x \) approaches a particular value gives insight into the function's behavior near that point. For the function \( f(x) \), as \( x \) approaches 0, the limit is calculated from the expression \( f(x) = x/x = 1 \). When \( x \to 0 \),
- The limit of 1 is still 1, because constant functions have the same value everywhere in their domain.
Continuity definition
The formal definition of continuity is crucial to determine whether a function is uninterrupted at a point \( x = a \). For a function \( f(x) \) to be continuous at \( x = a \), it must satisfy these three conditions:
- \( f(a) \) must be defined.
- The limit \( \lim_{{x \to a}} f(x) \) must exist.
- The value of the limit and the value of the function at \( a \) must be equal, i.e., \( \lim_{{x \to a}} f(x) = f(a) \).
Function behavior
Function behavior describes how the function acts over its domain. For this piecewise function, the behavior is straightforward:
- For any \( x eq 0 \), the function consistently equals 1, demonstrating constant behavior.
- At \( x = 0 \), it is explicitly defined as equal to 1.
Rational functions
Rational functions are composed as proportions of polynomials, expressed in the form \( \frac{P(x)}{Q(x)} \). The function \( f(x) \) simplifies to \( \frac{x}{x} \), which, once simplified, becomes a constant value of 1 for all \( x eq 0 \).Key aspects of rational functions include:
- They simplify polynomial expressions.
- They may demonstrate discontinuities at points where the denominator equals zero, potentially leading to undefined values or requiring additional definition, as seen at \( x = 0 \) in this piecewise function.
