Exercise · ZW-SEQUENTIAL-CONTINUITY-008 · ★★☆ Intermediate

The sequential criterion says that, for a real function \(f\) at a point \(c\) of its domain:

Choose one option.

Show answer
  • D. Being continuous and being sequentially continuous are equivalent

Explanation

If \(f\) were not continuous there would be an \(\varepsilon\) with no \(\delta\); taking \(\delta=1/n\) builds \(x_n\to c\) with \(|f(x_n)-f(c)|\ge\varepsilon\). Conversely, continuous implies sequential immediately.

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Review the concept: Sequential continuity

A function \(f\) is sequentially continuous at \(c\) if, for every sequence \((x_n)\) in the domain converging to \(c\), the sequence of images \(f(x_n)\) converges to \(f(c)\).

\[x_n\to c\ \Longrightarrow\ f(x_n)\to f(c)\]

More practice problems: sequential continuity

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