Exercise · ZW-UNIFORM-CONVERGENCE-007 · ★★☆ Intermediate
Why does \(x^n\) not converge uniformly on \([0,1]\)?
Choose one option.
Show answer
- B. Because its pointwise limit is discontinuous while each \(x^n\) is continuous.
Explanation
Each \(x^n\) is continuous on \([0,1]\), but the pointwise limit is \(0\) for \(x<1\) and \(1\) for \(x=1\), which is discontinuous. A uniform limit would be continuous, so the convergence is not uniform.
Review the concept: Uniform convergence
A sequence of functions \(f_n\) converges uniformly to \(f\) on a set \(A\) if, for every \(\varepsilon>0\), a single index \(N\) works for all points of \(A\).
\[\forall\varepsilon>0\ \exists N:\ n\ge N\Rightarrow|f_n(x)-f(x)|<\varepsilon\ \ \forall x\in A\]
More practice problems: uniform convergence
- ZW-UNIFORM-CONVERGENCE-001 Compute sup_x∈[0,1]|(x)/(n)-0|.
- ZW-UNIFORM-CONVERGENCE-002 Let f_n(x)=x^n on [0,1]. If f is its pointwise limit, what is f(1)?
- ZW-UNIFORM-CONVERGENCE-003 Compute sup_x∈[0,1)|x^n-0|.
- ZW-UNIFORM-CONVERGENCE-004 On which of these sets does x^n converge uniformly to 0?
- ZW-UNIFORM-CONVERGENCE-005 Let f_n(x)=(sin (nx))/(n) on mathbb R. What happens?
- ZW-UNIFORM-CONVERGENCE-006 The f_n are continuous on A and f_n→ f uniformly on A. Then f…
- ZW-UNIFORM-CONVERGENCE-008 Compute sup_x≥0(x)/(1+nx).
- ZW-UNIFORM-CONVERGENCE-009 Let f_n(x)=x^n(1-x) on [0,1], with pointwise limit 0. Does it converge uniformly?