Convergence of sequences and series
Lesson 7 of 10
Continuity and limits of functions
Objective. Test a function for continuity at a point.
The key idea
f is continuous at a if f(a) exists, the limit as x → a exists, and the two are equal. All three conditions matter.
Worked example
Is f(x) = (x² − 1)/(x − 1) continuous at x = 1?
- 1f(1) is undefined (0/0).
- 2The limit is 2 by cancelling.
- 3Since f(1) does not exist, it is not continuous.
Answer: No — a removable discontinuity
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Assuming a limit existing is enough for continuity.
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