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Chapter 1: Problem 57

Graph the indicated function. Find the interval(s) on which each function is continuous. $$ f(x)=\left\\{\begin{array}{ll} 1 & \text { if } 0 \leq x<1 \\ 2 & \text { if } 1 \leq x<2 \\ 3 & \text { if } 2 \leq x \leq 3 \end{array}\right. $$

Short Answer

Expert verified
The function is continuous on the intervals (0, 1), (1, 2), and [2, 3].

Step by step solution

01

Understand the Piecewise Function

The function is defined in three different sections. For each section, there's a constant value:- For the interval \(0 \leq x < 1\), \( f(x) = 1 \).- For the interval \(1 \leq x < 2\), \( f(x) = 2 \).- For the interval \(2 \leq x \leq 3\), \( f(x) = 3 \).
02

Graph Each Section

Plot each segment according to its corresponding interval and constant value:- From \(x = 0\) to \(x = 1\), draw a horizontal line at \(f(x) = 1\). Open circle at \(x = 1\).- From \(x = 1\) to \(x = 2\), draw a horizontal line at \(f(x) = 2\). Closed circle at \(x = 1\) and open circle at \(x = 2\).- From \(x = 2\) to \(x = 3\), draw a horizontal line at \(f(x) = 3\). Closed circles at both \(x = 2\) and \(x = 3\).
03

Determine Continuity at Intervals

Use the definition of continuity, which depends on the function being continuous at every point in the interval:- In the interval \((0, 1)\), the function is continuous as it is constant.- In the interval \([1, 2)\), the function is continuous as it is constant.- In the interval \([2, 3]\), the function is continuous as it is constant.
04

Analyze Points of Discontinuity

Check continuity at the endpoints and transition points:- At \(x = 1\), there is a change in value from 1 to 2, thus \(f\) is not continuous at \(x = 1\).- At \(x = 2\), there is a change in value from 2 to 3; however, the definition includes this point for the third interval with a closed circle. Hence, \(f\) is continuous at \(x = 2\).
05

List the Intervals of Continuity

Based on the continuity analysis, the continuous intervals are:- Open interval \((0, 1)\) because \(f\) is constant.- Open interval \((1, 2)\) because \(f\) is constant.- Closed interval \([2, 3]\) because \(f\) is constant and includes the endpoints.

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Key Concepts

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

Graphing Piecewise Functions
Graphing piecewise functions involves plotting different equations over specific intervals. These kinds of functions are defined by multiple sub-functions, each corresponding to a certain part of the domain.
In our example function, the graph is composed of three distinct parts:
  • From \(x = 0\) to \(x < 1\), the function value is 1, represented by a horizontal line.
  • From \(x = 1\) to \(x < 2\), the function value rises to 2, and is plotted as another horizontal line.
  • Finally, from \(x = 2\) to \(x = 3\), the value becomes 3, shown by the final horizontal line.
When graphing, particular attention is paid to endpoints. Open circles are used where a value is not included (e.g., at \(x = 1\) and \(x = 2\) for the first two intervals), while closed circles indicate inclusion, as at \(x = 2\) and \(x = 3\) for the last interval. This visual representation helps students understand different values the function takes throughout its domain.
Understanding Continuity Intervals
Continuity of a function means that it does not have any breaks, jumps, or holes within a given interval. For piecewise functions, assessing continuity involves looking at each segment separately.
Our example function is constant within its defined intervals, meaning it's continuous:
  • The interval \((0, 1)\) is continuous as \(f(x)\) remains constant at 1.
  • The interval \([1, 2)\) is continuous with \(f(x)\) staying at 2.
  • The interval \([2, 3]\) is continuous at a constant \(f(x) = 3\).
Despite changes in output value at the boundaries of different sub-functions, no gaps exist within these specific ranges. Each interval maintains its function's value without interruption, qualifying them as continuous.
Identifying Discontinuity Points
Discontinuity points are specific places where a function fails to be continuous. In piecewise functions, these often occur at the transition points between different segments.
For our piecewise function, let's examine each notable point:
  • At \(x = 1\), the value shifts from 1 to 2. This jump results in a discontinuity, marked by an open circle, showing that the function does not cover all values around this point.
  • At \(x = 2\), although the function shifts from 2 to 3, it starts a new segment that includes 2 as a valid point, hence it's not a discontinuity due to a closed circle marking inclusion.
Recognizing these points is crucial when analyzing piecewise functions, as they provide insight into the specific conditions and behaviors of the function across its domain.

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