Part BCSIR NET June 2024restriction-continuous-is-weaker

Restriction continuous is weaker

Let S be a dense subset of $R$ and $f : R \to R a$ given function. Define $g : S \to R$ by g(x) = f(x). Which of the following statements is necessarily true?

A.If f is continuous on the set S, then f is continuous on the set $R$ \ S
B.If g is continuous, then f is continuous on the set S
C.If g is identically 0 and f is continuous on the set $R$ \ S, then f is identically 0✓
D.If g is identically 0 and f is continuous on the set S, then f is identically 0

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Continuous on a bounded interval ⇒ bounded

Restriction continuous is weaker

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