Exercise · ZW-SEQUENTIAL-CONTINUITY-007 · ★★☆ Intermediate
Let \(f(x)=\sin(1/x)\) for \(x\ne0\) and \(f(0)=0\). With \(x_n=\dfrac{1}{\pi/2+2\pi n}\), what does \(f(x_n)\) converge to?
Choose one option.
Show answer
- C. \(1\)
Explanation
Here \(1/x_n=\pi/2+2\pi n\) and the sine of that equals \(1\). Since \(x_n\to0\) but \(f(x_n)=1\ne f(0)\), it is not continuous at \(0\).
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
- ZW-SEQUENTIAL-CONTINUITY-001 Let f(x)=x^2 and x_n=1+1/n. What does f(x_n) converge to?
- 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-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?
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