Questions: Sequential Characterization of Continuity

Question 1

A student wants to prove that f(x) = ⌊x⌋ (the floor function) is discontinuous at x = 2. Which approach is most direct using sequential continuity?

• A. Find ε > 0 and show no δ > 0 works by explicit case analysis
• B. Show that every sequence converging to 2 has f(xₙ) not converging to 2
• C. Exhibit a single sequence xₙ → 2 with f(xₙ) ↛ f(2)
• D. Show f is not differentiable at 2, which implies discontinuity

Answer: (C)

Sequential continuity makes discontinuity easy to demonstrate: one counterexample sequence suffices. For example, xₙ = 2 − 1/n → 2, but f(xₙ) = 1 for all n, so f(xₙ) → 1 ≠ 2 = f(2). You do not need to show all sequences fail — a single bad sequence establishes discontinuity. The ε-δ approach would work but requires more case analysis.

Question 2

Which statement correctly describes the relationship between sequential continuity and ε-δ continuity for real-valued functions on ℝ?

• A. ε-δ continuity implies sequential continuity, but not vice versa
• B. Sequential continuity implies ε-δ continuity, but not vice versa
• C. They are logically equivalent — each implies the other
• D. They agree on continuous functions but diverge on discontinuous ones

Answer: (C)

In ℝ (and all metric spaces), the two notions are exactly equivalent. This is the content of the sequential characterization theorem. A common misconception is that sequential continuity is weaker; it is not. Any discontinuity detectable by ε-δ is also detectable by a sequence, and vice versa. The equivalence holds in both directions.

Question 3

To prove a function is continuous at c using sequential continuity, it suffices to find one specific sequence (xₙ) converging to c such that f(xₙ) → f(c).

• T. True
• F. False

Answer: False

To prove continuity, you must show that EVERY sequence converging to c has its image converge to f(c). Finding one well-behaved sequence is not enough — a function could behave well on some sequences but fail on others. To prove DISCONTINUITY, however, finding a single bad sequence does suffice.

Question 4

The proof that sequential continuity implies ε-δ continuity is typically proved by contradiction: assume the function is not ε-δ continuous, then construct a sequence converging to c whose images do not converge to f(c).

• T. True
• F. False

Answer: True

The 'if' direction requires contradiction. Assuming f is not ε-δ continuous at c means: there exists ε > 0 such that for every δ > 0 = 1/n, some xₙ satisfies |xₙ − c| < 1/n but |f(xₙ) − f(c)| ≥ ε. This constructs a sequence xₙ → c with f(xₙ) ↛ f(c), contradicting the sequential hypothesis. This direction is the more surprising and instructive of the two.

Question 5

Explain in your own words why the sequential characterization of continuity is a valuable proof tool, even though it is logically equivalent to the ε-δ definition.

Model answer: The equivalence means you can choose whichever formulation is more convenient. Sequences are especially natural for disproving continuity (one counterexample sequence suffices), for Bolzano-Weierstrass arguments, and for reasoning about limits of compositions. The ε-δ formulation is often cleaner when constructing explicit bounds or proving continuity directly. The value is flexibility: the same mathematical content can be accessed in whichever form the problem makes most tractable.

Neither form is more powerful — they are equivalent. But a proof that is awkward in one language may be transparent in the other. Recognizing which mode fits the problem is itself a mathematical skill.