Science:Math Exam Resources/Courses/MATH102/December 2015/Question 14 (b)

MATH102 December 2015

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[hide] Question 14 (b)
At which values of $x$ is the function discontinuous?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Recall that a function $f(x)$ is continuous at $x=a$ if $\lim _{x\to a}f(x)$ exists, $f(a)$ is defined, and they are equal. If one of the assumptions is not satisfied, we say $f$ is not continuous at $x=a$ .

[show] Hint 2
A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil.

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[show] Solution
As in Hint 2, a function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. Since we can do this on the interval $(-\infty ,-2)$ , $(-2,-1)$ , $(-1,2)$ , and $(2,\infty )$ , the given function is continuous on the interval. Now we consider the points outside of the intervals: $x=-2,-1,2$ By part (a), the limits of $f$ at $x=\pm 2$ don't exist, so that $f$ is not continuous at $x=\pm 2$ . On the other hand, the function value at $f(-1)=3$ which is not equal to $\lim _{x\to -1}f(x)=1$ , therefore $f$ is also not continuous at $x=-1$ . Answer: $\color {blue}x=-2,-1,2$