A theorem assumes that f is continuous on [2, 5]. Its proof considers a case in which f is constant. Should being constant be added to the theorem's hypotheses?

• No. It is a temporary assumption for that proof case
• Yes. Every condition mentioned inside a proof is a theorem hypothesis
• Yes. The theorem applies only to the first case considered in its proof
• No. Theorem hypotheses never include conditions concerning functions

What should you extract from a mathematics paper on a first pass before checking its proof line by line?

• Every calculation in the first proof, with its intermediate steps
• The reference list and each author's institutional affiliation
• The statement of the main theorem and its hypotheses

Why try an example of a theorem before checking its proof in detail?

• It shows what the claim means in one case
• It proves the claim for every case the theorem allows
• It identifies every reference needed to justify the proof

A paper explicitly assumes readers know injective and defines short chain in the current passage. Injective is assumed background, and short chain is introduced there.

Circle one:   True   False

Section 5 uses tame cover and directs readers to Section 2 for its definition. How should your definition notes record this?

• Record it as undefined background, because only definitions on the current page count
• Record it as defined in Section 5, because mentioning a term supplies its meaning
• Record Section 2 as the referenced location of the term's definition
• Record it as newly invented, because a section reference establishes its historical origin

A lemma is cited exactly once in the main proof, to bound an error term. What is the lemma doing at that step?

• Restating the main theorem without providing the needed error estimate
• Supplying the error bound the main argument needs
• Listing every hypothesis of the paper without estimating the error

A theorem says: if f is a continuous real-valued function on the closed interval [0, 1], then f attains a maximum there. Which statement gives its hypotheses?

• f attains a maximum on [0, 1], and only one point attains that value
• f is differentiable on [0, 1], and its derivative vanishes at a maximum
• f is a continuous real-valued function on the closed interval [0, 1]

A theorem guarantees a maximum for a continuous real-valued function on a closed bounded interval. Your function is continuous and real-valued on (0, 1), excluding both endpoints. What follows from this theorem?

• The function has no maximum, because an omitted hypothesis makes the conclusion false
• The theorem does not settle this case; whether a maximum exists needs another argument
• The function must have a maximum, because continuity makes the interval endpoints irrelevant
• The function has exactly one maximum, because excluding endpoints removes every other candidate

You should postpone a definition needed to understand the main theorem until after reading its proof.

Circle one:   True   False

A theorem says every continuous real-valued f on [0, 3] has at least one maximum point. Which first-pass note records its hypotheses and conclusion without adding a claim?

• If f is continuous and real-valued on [0, 3], exactly one point attains its maximum value
• If f is continuous and real-valued on [0, 3], at least one maximum point exists
• If f is differentiable and real-valued on [0, 3], its only maximum point lies inside the interval