Questions: Sequential Characterization of Continuity

Question 1

You want to show that f(x) = sin(1/x) is discontinuous at x = 0 using the sequential characterization. Which approach is sufficient?

• A. Show that for every sequence xₙ → 0, the sequence f(xₙ) is bounded
• B. Find two sequences both converging to 0 that map to different limiting values under f
• C. Show that the epsilon-delta definition fails for every δ > 0
• D. Show that f(0) is undefined, so the sequential condition cannot be checked

Answer: (B)

The sequential characterization says f is continuous at c if and only if every sequence xₙ → c has f(xₙ) → f(c). To disprove continuity, you only need one 'bad' sequence — but finding two sequences converging to 0 with different image limits is the cleanest route here. Take xₙ = 1/(2πn): f(xₙ) = 0. Take yₙ = 1/(π/2 + 2πn): f(yₙ) = 1. Both converge to 0 but map to different limits, so f cannot be continuous there. Option D is a distractor: the sequential condition at c = 0 requires f(0) to be defined, but the discontinuity argument works independently of that.

Question 2

In the proof that epsilon-delta continuity implies the sequential condition (f continuous at c → xₙ → c implies f(xₙ) → f(c)), what role does the δ from continuity play?

• A. It sets the rate at which xₙ must converge to c
• B. It acts as the tolerance within which f(xₙ) must lie, bypassing the need for ε
• C. It provides a threshold: once xₙ is within δ of c, the continuity condition guarantees |f(xₙ) − f(c)| < ε, and the convergence of xₙ supplies an N beyond which this holds
• D. It replaces the ε in the definition of sequence convergence

Answer: (C)

The proof is a handoff between two definitions. Continuity at c (epsilon-delta) says: given ε > 0, there exists δ > 0 such that |x − c| < δ ⟹ |f(x) − f(c)| < ε. Sequence convergence says: there exists N such that for all n > N, |xₙ − c| < δ. Combining: for n > N, both conditions hold, so |f(xₙ) − f(c)| < ε. The δ is the bridge — it translates the 'closeness in domain' requirement of continuity into the 'eventually close enough' requirement of sequence convergence.

Question 3

The sequential characterization provides a definition of continuity that is strictly stronger than the epsilon-delta definition — functions can be epsilon-delta continuous without being sequentially continuous.

• T. True
• F. False

Answer: False

The sequential characterization is a theorem, not a new or stronger definition — it states an equivalence. A function f is epsilon-delta continuous at c if and only if it is sequentially continuous at c. The two formulations are interchangeable: any function that satisfies one satisfies the other, and any function that fails one fails the other. The sequential characterization is valuable precisely because it offers a different strategy for proofs, not because it imposes additional demands.

Question 4

To prove that a function f is discontinuous at c using the sequential characterization, it suffices to exhibit a single sequence xₙ → c for which f(xₙ) fails to converge to f(c).

• T. True
• F. False

Answer: True

The sequential condition for continuity requires that every sequence converging to c has its image converge to f(c). Negating a universal statement requires only a single counterexample. Therefore one carefully chosen sequence xₙ → c with f(xₙ) ↛ f(c) — whether because f(xₙ) diverges, converges to the wrong limit, or oscillates — is enough to conclude that f is discontinuous at c. This is far more economical than the contrapositive epsilon-delta argument.

Question 5

Why is the sequential characterization of continuity especially powerful for proving discontinuity, compared to a direct epsilon-delta argument?

Model answer: To disprove continuity via epsilon-delta, you must show that for some ε > 0, no δ > 0 works — a statement about all possible δ values, which requires constructing a counterexample for each. With the sequential characterization, disproving continuity requires only constructing one sequence that converges to c but whose image does not converge to f(c). Finding one specific sequence is much simpler than handling all δ, making discontinuity proofs cleaner and more intuitive.

The asymmetry is fundamental: continuous means every sequence works; discontinuous means some sequence fails. A single counterexample witnesses discontinuity directly. For oscillatory functions like sin(1/x), you can read the two sequences off the function's behavior — no epsilon-delta bookkeeping needed. The sequential approach effectively translates a statement about all sequences into a statement about one well-chosen sequence.