Continuity, uniform continuity, Lipschitz
Why this is asked: The chain Lipschitz ⇒ uniformly continuous ⇒ continuous, where each converse fails, and what compactness or boundedness of the domain changes. Know √x, 1/x, x², sin(1/x), sin(x²) by heart.
Continuity vs uniform continuity vs Lipschitz
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The trap here
“Continuous on a bounded interval ⇒ bounded” — false
f(x) = 1/x on (0,1)
Needs a compact (closed) domain.
Check yourself — select all that apply
Which of the following functions are uniformly continuous on the given domain?
Next: Differentiability, mean value theorems, Taylor, L'Hôpital
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Open this in the full syllabus view · Unit 1