A sharp maximum, an operator image and a grid certificate
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 3 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.
Work in the supremum norm on real continuous functions unless a task states otherwise.
State every common bound, compact-domain assumption and closedness argument.
A finite grid supplies a uniform certificate only with justified control between its nodes.
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Let F consist of real continuous functions on [0,1] satisfying |f(x)−f(y)|≤√|x−y| for all x,y and ∫₀¹f(t)dt=0. Prove F is nonempty and compact in the supremum norm. Prove f↦f(1) attains a maximum. Then find that maximum and prove that its maximizer is unique. An existence theorem alone does not determine the maximizing function.
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.
For real f∈C([0,1]) with ||f||∞≤1 define (Tf)(x)=∫₀¹|x−t|f(t)dt. Prove the image family is relatively compact in C([0,1]). Is the image itself compact? To settle the second question, use f_n(t)=max(−1,min(1,n(t−1/2))) for integers n≥2 and compare with the step s(t)=−1 for t<1/2 and 1 for t>1/2 (choose either value at 1/2). Prove uniform convergence of Tf_n to a function outside the image; do not simply assume a continuous image of a bounded set is compact.
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.
Two functions on [0,1] have Lipschitz constants at most 3 and 1 respectively. At the nodes S={0,1/5,3/5,1}, their discrepancies are at most 1/50. Find the sharp uniform error certificate based only on these facts, and construct functions attaining it. Then determine the smallest number m of equal subintervals that would guarantee uniform discrepancy at most 1/10 with the same node error. Explain what fails if each function is merely continuous with no specified common modulus.
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 F consist of real continuous functions on [0,1] satisfying |f(x)−f(y)|≤√|x−y| for all x,y and ∫₀¹f(t)dt=0. Prove F is nonempty and compact in the supremum norm. Prove f↦f(1) attains a maximum. Then find that maximum and prove that its maximizer is unique. An existence theorem alone does not determine the maximizing function.
For real f∈C([0,1]) with ||f||∞≤1 define (Tf)(x)=∫₀¹|x−t|f(t)dt. Prove the image family is relatively compact in C([0,1]). Is the image itself compact? To settle the second question, use f_n(t)=max(−1,min(1,n(t−1/2))) for integers n≥2 and compare with the step s(t)=−1 for t<1/2 and 1 for t>1/2 (choose either value at 1/2). Prove uniform convergence of Tf_n to a function outside the image; do not simply assume a continuous image of a bounded set is compact.
Two functions on [0,1] have Lipschitz constants at most 3 and 1 respectively. At the nodes S={0,1/5,3/5,1}, their discrepancies are at most 1/50. Find the sharp uniform error certificate based only on these facts, and construct functions attaining it. Then determine the smallest number m of equal subintervals that would guarantee uniform discrepancy at most 1/10 with the same node error. Explain what fails if each function is merely continuous with no specified common modulus.
