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Chapter 1: Problem 32

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\left\\{\begin{array}{ll} \csc \frac{\pi x}{6}, & |x-3| \leq 2 \\ 2, & |x-3|>2 \end{array}\right. $$

Short Answer

Expert verified
The function \(f(x)\) is not continuous at \(x = 3\), and this discontinuity is not removable.

Step by step solution

01

Analyze the Discontinuities of the Cosecant Function

Let's first examine the function \(\csc(\frac{\pi x}{6})\). This is f(x) when \( |x-3| \leq 2 \), which corresponds to the interval \( 1 \leq x \leq 5 \). The cosecant function has discontinuities whenever its argument is a multiple of \(\pi \). Thus, the function \(\csc(\frac{\pi x}{6})\) will be not continuous whenever \(\frac{\pi x}{6}\) is a multiple of \(\pi \), specifically when \(x\) is an integer multiple of 6. However, between \( 1 \leq x \leq 5 \), there are no multiple of 6. Therefore, within this interval, there are no inherent discontinuities due to the cosecant function.
02

Make Sure the Constant Function is Always Continuous

The function 2 is f(x) when \( |x-3| > 2 \), which corresponds to \( x < 1 \) and \( x > 5 \). This function is always continuous as it is a constant function.
03

Check Continuity at Boundary points

The boundary point is at \( x = 3 \). At this point, the function is defined to be \(\csc(\frac{\pi x}{6})\), which gives \(\csc(\frac{\pi(3)}{6}) = \csc(\frac{\pi}{2}) = 1\). However, as we approach \(x = 3\) from values larger than 3 (from the second case), the function approaches 2, not 1. Therefore, \(x = 3\) is a discontinuity, and it is not removable as the function does not approach the same value from both directions.

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Most popular questions from this chapter

Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{3}+x-1 $$

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\operatorname{arcsec} 2 x $$

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arctan x+\frac{\pi}{2} $$

Determine conditions on the constants \(a, b,\) and \(c\) such that the graph of \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\).

$$ \lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\\{\begin{array}{ll} 3, & x \leq 2 \\ 0, & x>2 \end{array}\right. $$
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