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Chapter 4: Problem 27

Prove that the function \(f(x)=\left\\{\begin{array}{l}2-x: x \geq 1 \\ x+2: x<1\end{array}\right.\) is discontinuous at \(x=1\).

Short Answer

Expert verified
The function \(f(x)\) is discontinuous at \(x=1\) because the left-hand limit (3) is not equal to the right-hand limit (1).

Step by step solution

01

Analyze the function for \(x \geq 1\)

For \(x \geq 1\), the function \(f(x)\) is defined as \(2-x\). Let's calculate the limit of \(f(x)\) as x approaches one from the positive side (right-hand limit): \(\lim_{x\to 1^+} f(x)=\lim_{x\to 1^+} (2-x)\). As x gets close to 1, \(2-x\) will equal 1.
02

Analyze the function for \(x < 1\)

For \(x < 1\), the function \(f(x)\) is defined as \(x+2\). Let's calculate the limit of \(f(x)\) as x approaches one from the negative side (left-hand limit): \(\lim_{x\to 1^-} f(x)=\lim_{x\to 1^-} (x+2)\). As x gets close to 1, \(x+2\) will equal 3.
03

Compare the left-hand limit, right-hand limit and the function value at \(x=1\)

We compare the the right-hand limit as \(x\to 1^+\), left-hand limit as \(x\to 1^-\) and the function value at \(x=1\). On comparing, we find that the left-hand limit which equals to 3 is not equal to the right-hand limit which equals to 1. Also, the value of function \(f(x)\) at \(x=1\) (which equals to 1 as defined), is not equal to the left-hand limit. Hence, the function is discontinuous at \(x=1\).

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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 a function composed of multiple line segments or curves, each of which is defined over a different interval of the domain. In the exercise above, the function is written as:
  • \(f(x) = 2-x\) for \(x \geq 1\)
  • \(f(x) = x+2\) for \(x < 1\)
This implies that there are two separate expressions which define different parts of the function based on the value of \(x\). Each piece is valid for a specific interval. This way of defining a function allows it to behave differently over distinct sections of its domain. It's important to consider such characteristics, as they can lead to discontinuities, especially at the points where definitions change.
Limits
Limits help us understand the behavior of functions as they approach a specific point. We use limits to determine what value a function approaches, even if it is not actually defined at that point. Consider the function from our exercise:
  • By calculating \(\lim_{x\to 1^+} (2-x)\), we find that as \(x\) is approaching 1 from the right (getting closer to 1 but more than 1), the value approaches 1.
  • For \(\lim_{x\to 1^-} (x+2)\), \(x\) approaches 1 from the left (getting closer to 1 but less than 1), and the value approaches 3.
Limits give us crucial insight into how a function behaves around points of interest, like potential discontinuities.
Continuity
Continuity refers to when a function runs smoothly without any breaks, holes, or jumps. A function is continuous at a point \(x = a\) if:
  • The function \(f(x)\) is defined at \(x = a\)
  • The limit as \(x\) approaches \(a\) from both directions exists and is equal
  • The limit is equal to the function's value \(f(a)\)
For the exercise's piecewise function:- At \(x = 1\), the left-hand and right-hand limits do not match (3 \(eq\) 1)- The function value at 1, which is defined to be 1, doesn't match the left-hand limit of 3
This mismatch indicates that the function is discontinuous at \(x = 1\). Continuity ensures a seamless journey across a function's graph.
Right-hand Limit
Right-hand limit examines the behavior of a function as it approaches a specific value from the right. Represented as \(\lim_{x \to a^+} f(x)\), it focuses on values of \(x\) that are greater than \(a\) but approach it. In our piecewise function example, we specifically analyzed the right-hand limit at \(x = 1\):
  • For the expression \(2-x\) when \(x \geq 1\)
  • The right-hand limit \(\lim_{x \to 1^+} (2-x)\) is 1
This means as \(x\) approaches 1 from the right, the function \(f(x)\) values head towards 1. Understanding the right-hand limit helps in identifying if a function maintains continuity from this side.
Left-hand Limit
The left-hand limit observes how a function behaves as it approaches a particular point from the left. Denoted \(\lim_{x \to a^-} f(x)\), it considers values less than \(a\) but nearing it. Examining the left-hand limit in our problem:
  • For \(x + 2\) when \(x < 1\)
  • The left-hand limit \(\lim_{x \to 1^-} (x + 2)\) equals 3
This reveals that approaching 1 from the left, the function \(f(x)\) tends towards 3. The disparity between left and right-hand limits at a point is central to understanding discontinuity, as differing limits indicate a break in the function at that point.

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Most popular questions from this chapter

Let \(f(x)=\lim _{n \rightarrow \infty}\left(\frac{|a+\sin \pi x|^{n}-1}{|a+\sin \pi x|^{n}+1}\right), x \in(0,6)\) Find the number of discontinuity, when \(a=0\) and \(L\).

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