MATH110 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 • Q8 • Q9 • Q10 (a) • Q10 (b) • QS01 10(a) • QS01 10(b) •
Question 01 (c)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample.
c) is continuous at all real numbers.
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
You need to check three things:
1. That the function is continue at the break point given by
2. That the right hand side of the function is continuous at all points larger than 1.
3. That the left hand side of the function is continuous at all points smaller than 1.
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False: To hunt for discontinuities, we have to check inside each piece and also at the boundary.
For the boundary to be continuous, we must have:
So the limits exists and is equal to which happens to be . So the function in continuous at the boundary.
On the right of , the function is continuous since is only undefined for which is not covered by this case.
On the left of , the function is discontinuous at since the denominator is . This fall inside the region considered. So is discontinuous at .
That means is NOT continuous over all real numbers.
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