Questions: Epsilon-Delta Continuity

Question 1

Which statement correctly captures the ε-δ definition of continuity of f at c?

• A. For some ε > 0, for all δ > 0, |x − c| < δ implies |f(x) − f(c)| < ε
• B. For all ε > 0, there exists δ > 0 such that |x − c| < δ implies |f(x) − f(c)| < ε
• C. For all δ > 0, there exists ε > 0 such that |f(x) − f(c)| < ε implies |x − c| < δ
• D. For all ε > 0, for all δ > 0, |x − c| < δ implies |f(x) − f(c)| < ε

Answer: (B)

The correct quantifier order is 'for all ε, there exists δ' — ε is the adversarial challenge, δ is the response. Option A fails because 'for some ε' allows cherry-picking an easy ε rather than handling all of them. Option C reverses the implication direction (output tolerance implying input restriction). Option D claims one δ works for every ε simultaneously, which is a different (stronger) condition.

Question 2

To prove f is continuous at c using the ε-δ definition, it suffices to exhibit one specific pair (ε₀, δ₀) that satisfies the condition.

• T. True
• F. False

Answer: False

Continuity requires finding a valid δ for EVERY ε > 0, no matter how small. Verifying the condition for a single ε₀ says nothing about other values of ε. A complete proof must provide δ as a function of ε — a strategy that works universally. The same logic applies to sequence convergence: finding one N for one ε does not prove the sequence converges.

Question 3

What does the ε-δ definition formalize that the informal notion of 'no jumps or holes' does not?

Model answer: The ε-δ definition gives a quantitative, provable criterion: for any output tolerance ε, there is an input restriction δ that guarantees the output stays within ε. This handles pathological functions and non-geometric domains where 'no jumps' is ambiguous, and it produces checkable proofs.

Informal geometric descriptions fail for exotic functions (Dirichlet-type functions, functions on abstract metric spaces) where drawing a graph is impossible or misleading. The ε-δ formulation is domain-agnostic and admits rigorous proof by construction of δ from ε.