- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Continuity practice question 1
A function is continuous at , and . Which of the following must be true?
Test each statement against the definition of continuity at . What, exactly, does that definition promise?
By definition, is continuous at when . Put the given limit into that equation with .
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Continuity at a point is one equation: the limit equals the value. Given the limit and given continuity, the value is forced.
Working
The definition.
Substitute the given limit.
Answer
The answer is f of 2 equals 5.
Check
f(x) = 5 + |x − 2| is continuous with limit 5 at 2 and has f(2) = 5, yet it is not 5 near 2 and has no f′(2).
Where this goes wrong
Choosing “f is constant near 2”. A limit of 5 says the values get close to 5, not that they equal 5. Continuity adds nothing about differentiability either.
Leans on
The definition of a limit, from Limits
Try a similar question
Find the value of that makes continuous at .Saved in this browser only.