Calculus 1 — 7.3: Continuity on Intervals

Описание: How do you prove a function is continuous at the endpoints of its domain, where two-sided limits don't exist? This lesson formalizes continuity on open, closed, and half-open intervals using one-sided limits, then walks through a complete proof that f(x) = √(16 − x²) is continuous on [−4, 4].

Key concepts covered:
• Why two-sided limit definitions break down at domain boundaries
• Continuity on open intervals: every interior point satisfies the standard limit definition
• One-sided limits: right-hand (x → c⁺) and left-hand (x → c⁻) notation and meaning
• Continuity from the right and continuity from the left at a point
• The three-part definition of continuity on a closed interval [a, b]: interior two-sided limits, right-hand limit at the left endpoint, left-hand limit at the right endpoint
• Why checking only one direction at endpoints is mathematically sufficient
• Continuity rules for half-open intervals [a, b) and (a, b]: closed endpoints require one-sided limit checks, open endpoints are skipped
• Full worked proof: showing √(16 − x²) is continuous on [−4, 4] by verifying interior continuity, the right-hand limit at x = −4, and the left-hand limit at x = 4
• The mnemonic: approach from where the interval exists

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 1.4:  Continuity of Fun...