Exercise · ZW-SEQUENTIAL-CONTINUITY-006 · ★★☆ Intermediate
To prove with the sequential criterion that \(f\) is not continuous at \(c\), it suffices to exhibit:
Choose one option.
A A sequence \(x_n\to c\) such that \(f(x_n)\) does not converge to \(f(c)\) B A sequence \(x_n\to c\) such that \(f(x_n)\to f(c)\) C A sequence \(x_n\) that does not converge D Two different sequences converging to \(c\) Show answer A. A sequence \(x_n\to c\) such that \(f(x_n)\) does not converge to \(f(c)\)← Back to the question
Explanation A single counterexample (a sequence converging to \(c\) whose images do not converge to \(f(c)\)) refutes the condition «for every sequence».
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