- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Continuity practice question 4
On which set is the function below continuous?
Here is the natural logarithm, defined only for positive inputs. A quotient of continuous functions is continuous wherever it is defined, so list what the root, the logarithm and the denominator each need, and keep the points that pass all three.
The root needs , where 0 is allowed. The logarithm needs , and as a denominator it needs .
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Roots, logarithms and quotients are continuous wherever they are defined, so f is continuous exactly on its domain. Find the domain one condition at a time.
Working
A square root needs a non-negative input, and 0 is allowed.
A logarithm needs a strictly positive input.
The logarithm sits in the denominator, and log 1 = 0.
Keep what passes all three. At x = 5, f(5) = 0/log 5 = 0 and f is continuous from the left, so 5 belongs.
Answer
The answer is every x with 0 < x ≤ 5 except x = 1.
Check
f(5) = 0 and f(4.99) ≈ 0.06, so 5 belongs, while near x = 1 the quotient blows up because log 1 = 0.
Where this goes wrong
Keeping x = 1 or dropping x = 5. The logarithm is 0 at x = 1, which puts 0 in the denominator; the root is 0 at x = 5, which is harmless because it sits in the numerator.
Leans on
Domain of a logarithm, from Functions, graphs and inverses
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