Continuity practice question 2
On which set is the function below continuous?
A sum of continuous functions is continuous wherever both terms are defined. Find where is defined, find where is defined, and keep the overlap.
A logarithm needs a strictly positive input, so . A denominator cannot be 0, so .
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Logarithms and rational functions are continuous wherever they are defined, so the sum is continuous exactly where both terms are defined.
Working
A logarithm needs a strictly positive input; 0 is not allowed.
A denominator cannot be zero.
Keep the points that pass both tests: the interval (−2, 2) with x = 1 removed.
Answer
The answer is every x strictly between −2 and 2 except x = 1.
Check
f(1.5) = log 1.75 + 2 ≈ 2.56 is defined, while f(2.5) would need log(−2.25) and f(1) would divide by 0.
Where this goes wrong
Including the endpoints ±2, or forgetting the gap at 1. log 0 is undefined, so the logarithm needs 4 − x² strictly positive, and the second term removes x = 1 from the middle of that interval.
The idea behind it
The continuity lesson, a figure to move, in a few steps
Leans on
Domain of a logarithm, from Functions, graphs and inverses
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