Let \(f(x)=x^2\) and \(x_n=1+1/n\). What does \(f(x_n)\) converge to?

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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 \(f(c)\).

More practice problems: sequential continuity

• ZW-SEQUENTIAL-CONTINUITY-002 Let f(x)=1 if x≥0 and f(x)=-1 if x<0. Which sequence converging to 0 shows that f is not continuous…
• ZW-SEQUENTIAL-CONTINUITY-003 Let f be the Dirichlet function (1 at rationals, 0 at irrationals) and x_n=√2/n. What is lim f(x_n)?
• ZW-SEQUENTIAL-CONTINUITY-004 If f is continuous at c, what holds?
• ZW-SEQUENTIAL-CONTINUITY-005 Let f(x)=(x^2-1)/(x-1) if xne1 and f(1)=3. With x_n=1+1/n, what does f(x_n) converge to?
• ZW-SEQUENTIAL-CONTINUITY-006 To prove with the sequential criterion that f is not continuous at c, it suffices to exhibit:
• ZW-SEQUENTIAL-CONTINUITY-007 Let f(x)=sin (1/x) for xne0 and f(0)=0. With x_n=(1)/(π/2+2π n), what does f(x_n) converge to?
• ZW-SEQUENTIAL-CONTINUITY-008 The sequential criterion says that, for a real function f at a point c of its domain:
• ZW-SEQUENTIAL-CONTINUITY-009 Let f(x)=x if x is rational and f(x)=0 if irrational. At which points is it continuous?

ZW-SEQUENTIAL-CONTINUITY-002 →