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Is this true?

Pointwise limit of continuous functions is continuous

No — it is false.

The counterexample

This one comes with the Notes pack.

See pricing

85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Watch it happen

Watch uniform convergence fail: xⁿ on [0, 1]

uniform convergence

Others that fail the same way

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

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