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.
Use neighborhoods relative to [0,1], including at the endpoints.
A null set must have interval covers of arbitrarily small total length; density and cardinality alone do not decide this.
Prove each infinite-tail or covering claim. A finite drawing is supporting intuition, not the proof.
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Enumerate the rationals in (0,1) without repetition as r₁,r₂,…, and define F(x)=Σ from j=1 to ∞ of 2^(−j)·1{x≥r_j} on [0,1]. Prove the series defines a bounded increasing function. Determine every point of discontinuity and its jump, including endpoint behavior. Prove F is right-continuous. Derive its Riemann integral as an infinite series with an explicit finite-sum error bound. Finally replace F(r_j) by the midpoint of its one-sided limits at each rational r_j. Does monotonicity or the integral change?
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.
Start with C₀=[0,1]. From every closed component of C_n remove its open middle third to obtain C_(n+1), and put C=intersection over n≥0 of C_n. Prove C is closed and null, but uncountable. Determine the relative point oscillation of f=1_C on [0,1], including at 0 and 1, and prove its Riemann integral. Explain why the rational indicator on the same interval has a different integrability conclusion even though the rationals are countable and null.
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 bounded functions f and g on [0,1], prove the point-oscillation inequality ω_(f+g)(x)≤ω_f(x)+ω_g(x). Deduce the corresponding containment of discontinuity sets and a proof that a sum of two Riemann-integrable functions is integrable. Is the containment always equality? Test f(x)=1_Q(x) and g(x)=x−1_Q(x), where Q denotes the rationals. Compute all three point oscillations and explain what these examples do and do not imply about cancellation and changes on null sets.
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
Enumerate the rationals in (0,1) without repetition as r₁,r₂,…, and define F(x)=Σ from j=1 to ∞ of 2^(−j)·1{x≥r_j} on [0,1]. Prove the series defines a bounded increasing function. Determine every point of discontinuity and its jump, including endpoint behavior. Prove F is right-continuous. Derive its Riemann integral as an infinite series with an explicit finite-sum error bound. Finally replace F(r_j) by the midpoint of its one-sided limits at each rational r_j. Does monotonicity or the integral change?
Start with C₀=[0,1]. From every closed component of C_n remove its open middle third to obtain C_(n+1), and put C=intersection over n≥0 of C_n. Prove C is closed and null, but uncountable. Determine the relative point oscillation of f=1_C on [0,1], including at 0 and 1, and prove its Riemann integral. Explain why the rational indicator on the same interval has a different integrability conclusion even though the rationals are countable and null.
For bounded functions f and g on [0,1], prove the point-oscillation inequality ω_(f+g)(x)≤ω_f(x)+ω_g(x). Deduce the corresponding containment of discontinuity sets and a proof that a sum of two Riemann-integrable functions is integrable. Is the containment always equality? Test f(x)=1_Q(x) and g(x)=x−1_Q(x), where Q denotes the rationals. Compute all three point oscillations and explain what these examples do and do not imply about cancellation and changes on null sets.
