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

Restriction continuous is weaker

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

  1. A.If f is continuous on the set S, then f is continuous on the set R\mathbb{R} \ S
  2. B.If g is continuous, then f is continuous on the set S
  3. C.If g is identically 0 and f is continuous on the set R\mathbb{R} \ S, then f is identically 0✓ Correct
  4. 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, and why this one tests property not inherited.

See pricing

50 are analysed free — try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: Continuous on a bounded interval ⇒ bounded

More on this topic

The chapter behind this: Continuity vs uniform continuity vs Lipschitz — free to read

From Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

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