- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Continuity practice question 7
The graph of a function is shown. Only the value may be changed; every other value stays as it is. What must be changed to so that is continuous at ?
Continuity at means . Changing moves the filled point but not the curve around it.
Follow the curve towards from each side. What height does it approach?
A whole number.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Changing f(−1) cannot change the limit, which depends only on the values near −1. So read the limit from the graph, then make the value equal to it.
Working
From both sides the curve approaches the open circle at height 3.
The limit exists, but the filled point puts the value at 1, so f is not continuous at −1 as drawn.
Moving the filled point up into the open circle fills the hole: the discontinuity is removable.
Answer
The answer is f(−1) must be changed to 3.
Where this goes wrong
Answering 1, the value f has now. The filled point is the value that breaks continuity; the limit, 3, is set by the curve on either side, and the value must match it.
Leans on
Two-sided limits, from Limits
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