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Is this true?

Continuous on a bounded interval ⇒ bounded

No — it is false.

The counterexample

f(x) = 1/x on (0,1)

Needs a compact (closed) domain.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Watch it happen

The ε–δ limit machine

continuity

Others that fail the same way

From Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

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