Exercise · ZW-SEQUENTIAL-CONTINUITY-009 · ★★★ Advanced
Let \(f(x)=x\) if \(x\) is rational and \(f(x)=0\) if irrational. At which points is it continuous?
Choose one option.
Show answer
- B. Only at \(0\)
Explanation
At \(c\ne0\), a sequence of rationals tending to \(c\) gives \(c\) and one of irrationals gives \(0\ne c\). At \(0\), \(|f(x_n)|\le|x_n|\to0=f(0)\).
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-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:
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