Analysis II: uniform control, spaces and local theorems
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 14 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose a method and write the argument before opening guidance. State the field, domain, dimensions and theorem hypotheses. A computed example does not prove a general assertion.
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Let U be the plane with the two vertical lines x=−1 and x=1 removed, and H(x,y)=(x²y, sin(x)+y). A differentiable G has DG=DH on U and G(0,0)=H(0,0). Must G=H everywhere? Determine the necessary component anchors. Then replace U by the punctured plane. Finally, if ||DG−DH||≤ε instead of equality, what bound follows along a domain-valid polygonal route of length L from an anchor with discrepancy at most δ?
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let u=x−1, v=y+2 and G(x,y)=7+u²+v²+5uv(u²−v²)/(u²+v²), taking the last term as zero at u=v=0. Compute the gradient and all second partials at (1,−2). A proposed solution averages the mixed entries and uses the positive matrix 2I to claim a minimum. Diagnose the error and classify the point by explicit paths.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Define K(x)=x²sin(1/x²) for x≠0 and K(0)=0 on [−1,1], and fₙ(x)=K(x)+sin(x)/n. Prove uniform convergence of the functions and derivatives, identify the derivative of the limit, and show that it is unbounded near zero. Explain why this does not contradict the derivative-interchange theorem and why the C¹/continuous-integral proof is unavailable.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct ψ(x)=Σ from n=0 to infinity of (−1)ⁿx³ⁿ/n! without assuming its closed form. Prove ψ′=−3x²ψ and ψ(0)=1. For the truncation through N=3 on[−1,1], derive value and derivative tail bounds and compute the ODE residual at x=1. Finally give a justified series for ∫₋₁¹ψ(x)dx.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For fₙ(x)=xⁿ/(1+xⁿ) on [0,1], find the pointwise limit and supremum error, including whether it is attained. Prove uniform convergence on [0,1/2]. State the uniform Cauchy criterion and explain its equivalence to uniform convergence for real-valued functions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On [0,1], let fₙ(x)=n max(1−n|x−1/n|,0), n≥2. Find its pointwise limit, maximum and integral. Use it to distinguish pointwise from uniform limit theorems, then prove the uniform integral-interchange theorem for Riemann-integrable functions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show that fₙ(x)=arctan(nx)/n converges uniformly on [−1,1], but its derivatives cannot converge pointwise to the derivative of the limit. Give a sufficient derivative-interchange theorem and prove it for C¹ functions on a compact interval.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Analyze Σ from n=1 to infinity xⁿ/n² on [−1,1]. Prove uniform convergence, justify termwise integration, and determine where termwise differentiation is justified. Explain the obstruction at x=1.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare the usual metric on R with d(x,y)=min(1,|x−y|) and with the discrete metric. Prove that the first two have the same convergent and Cauchy sequences. Determine the continuity directions of the identity maps between discrete and usual R.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Give a bounded complete metric space that is not compact and a totally bounded space that is not complete. Prove the relevant failures directly, and explain the role of completeness plus total boundedness in the metric compactness theorem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that a continuous function R→Z, with both carrying their usual subspace topologies, is constant. Exhibit a separation showing Q is disconnected. Explain why a picture containing many nearby points is not evidence of connectedness.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For T(x)=(x+3/x)/2 on [3/2,2], prove the self-map and contraction hypotheses. Starting at x₀=2, compute x₁ and an a posteriori bound on its distance to the fixed point. Explain why the same theorem cannot be invoked on all positive real numbers with this estimate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For F(x,y)=(eˣ,yeˣ), compute DF and its Euclidean operator norm at (0,1). Give the second-order Taylor polynomial at (0,0) with a justified local remainder order. For g(u,v)=u²+v², check the derivative of g∘F by the chain rule and directly.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For H(u,v)=(u+v,uv), prove local invertibility near (2,3) and compute the derivative of its inverse at (5,6). Explain why this is not global injectivity. For x²+xy+2y²=11 near (1,2), justify a differentiable branch y(x) and find its derivative at 1.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let U be the plane with the two vertical lines x=−1 and x=1 removed, and H(x,y)=(x²y, sin(x)+y). A differentiable G has DG=DH on U and G(0,0)=H(0,0). Must G=H everywhere? Determine the necessary component anchors. Then replace U by the punctured plane. Finally, if ||DG−DH||≤ε instead of equality, what bound follows along a domain-valid polygonal route of length L from an anchor with discrepancy at most δ?
Let u=x−1, v=y+2 and G(x,y)=7+u²+v²+5uv(u²−v²)/(u²+v²), taking the last term as zero at u=v=0. Compute the gradient and all second partials at (1,−2). A proposed solution averages the mixed entries and uses the positive matrix 2I to claim a minimum. Diagnose the error and classify the point by explicit paths.
Define K(x)=x²sin(1/x²) for x≠0 and K(0)=0 on [−1,1], and fₙ(x)=K(x)+sin(x)/n. Prove uniform convergence of the functions and derivatives, identify the derivative of the limit, and show that it is unbounded near zero. Explain why this does not contradict the derivative-interchange theorem and why the C¹/continuous-integral proof is unavailable.
Construct ψ(x)=Σ from n=0 to infinity of (−1)ⁿx³ⁿ/n! without assuming its closed form. Prove ψ′=−3x²ψ and ψ(0)=1. For the truncation through N=3 on[−1,1], derive value and derivative tail bounds and compute the ODE residual at x=1. Finally give a justified series for ∫₋₁¹ψ(x)dx.
For fₙ(x)=xⁿ/(1+xⁿ) on [0,1], find the pointwise limit and supremum error, including whether it is attained. Prove uniform convergence on [0,1/2]. State the uniform Cauchy criterion and explain its equivalence to uniform convergence for real-valued functions.
On [0,1], let fₙ(x)=n max(1−n|x−1/n|,0), n≥2. Find its pointwise limit, maximum and integral. Use it to distinguish pointwise from uniform limit theorems, then prove the uniform integral-interchange theorem for Riemann-integrable functions.
Show that fₙ(x)=arctan(nx)/n converges uniformly on [−1,1], but its derivatives cannot converge pointwise to the derivative of the limit. Give a sufficient derivative-interchange theorem and prove it for C¹ functions on a compact interval.
Analyze Σ from n=1 to infinity xⁿ/n² on [−1,1]. Prove uniform convergence, justify termwise integration, and determine where termwise differentiation is justified. Explain the obstruction at x=1.
Compare the usual metric on R with d(x,y)=min(1,|x−y|) and with the discrete metric. Prove that the first two have the same convergent and Cauchy sequences. Determine the continuity directions of the identity maps between discrete and usual R.
Give a bounded complete metric space that is not compact and a totally bounded space that is not complete. Prove the relevant failures directly, and explain the role of completeness plus total boundedness in the metric compactness theorem.
Prove that a continuous function R→Z, with both carrying their usual subspace topologies, is constant. Exhibit a separation showing Q is disconnected. Explain why a picture containing many nearby points is not evidence of connectedness.
For T(x)=(x+3/x)/2 on [3/2,2], prove the self-map and contraction hypotheses. Starting at x₀=2, compute x₁ and an a posteriori bound on its distance to the fixed point. Explain why the same theorem cannot be invoked on all positive real numbers with this estimate.
For F(x,y)=(eˣ,yeˣ), compute DF and its Euclidean operator norm at (0,1). Give the second-order Taylor polynomial at (0,0) with a justified local remainder order. For g(u,v)=u²+v², check the derivative of g∘F by the chain rule and directly.
For H(u,v)=(u+v,uv), prove local invertibility near (2,3) and compute the derivative of its inverse at (5,6). Explain why this is not global injectivity. For x²+xy+2y²=11 near (1,2), justify a differentiable branch y(x) and find its derivative at 1.
