- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Continuity practice question 3
On which set is continuous?
A quotient of continuous functions is continuous wherever it is defined. Where is defined?
The root needs , and because it sits in a denominator it cannot be 0 either, so you need . Which , positive and negative, satisfy that?
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Square roots need a non-negative argument, and a denominator cannot be zero. The function is continuous everywhere it is defined.
Working
Strictly positive: zero would put 0 in the denominator.
Open at ±2 because the endpoints are excluded.
Answer
The answer is all x with absolute value greater than 2.
Check
g(3) = g(−3) ≈ 0.45, but g(0) would need √(−4), and g blows up as x → 2⁺ or −2⁻.
Where this goes wrong
Including the endpoints ±2. The square root is fine at 0, but it sits in a denominator, so 0 is not allowed.
Leans on
Domain of a square root, from Functions, graphs and inverses
Try a similar question
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