Continuity, the Intermediate Value Theorem, and the Extreme Value Theorem

Most functions that appear in finance — option prices, discount factors, probability density functions — are continuous. Continuity is not merely a nicety: it is the hypothesis that allows you to evaluate limits by substitution, guarantees that Black-Scholes prices vary smoothly with inputs, and underpins the two results that follow — the Intermediate Value Theorem and the Extreme Value Theorem. The IVT guarantees existence of roots (and hence implied volatility), while the EVT guarantees that a portfolio optimisation on a closed, bounded domain has a solution.

• state the three conditions for continuity at a point and check each;
• identify removable, jump, and infinite discontinuities by examining the limit;
• state and verify continuity on an interval; list which function families are continuous on their natural domains;
• apply the IVT to prove a root exists in a given interval;
• state the EVT and explain why it requires both a closed interval and continuity.

1. Continuity at a point
2. Three types of discontinuity
3. Continuity on an interval
4. The Intermediate Value Theorem
5. The Extreme Value Theorem