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This lecture completes the foundations by defining continuity rigorously, exploring its consequences, and introducing the formal epsilon-delta definition of limits.

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📋 At a Glance

• [ ] Define continuity at a point using three conditions
• [ ] Test functions for continuity at specific points
• [ ] Identify and classify discontinuities (removable, jump, essential)
• [ ] Understand the Intermediate Value Theorem and its applications
• [ ] Know which standard functions are continuous and where
• [ ] Understand continuity of composite functions
• [ ] Grasp the epsilon-delta definition of limits (conceptually)
• [ ] Apply continuity to find limits of composite expressions

1. Continuity

Intuitive Definition

A function is continuous if you can draw its graph without lifting your pen.

Formal Definition

A function f is continuous at x = a if all three conditions hold:

1. f(a) is defined          (the function exists at a)
2. lim_{x→a} f(x) exists     (the limit exists)
3. lim_{x→a} f(x) = f(a)     (the limit equals the function value)

A function is continuous on an interval if it's continuous at every point in that interval.

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Key Insight: Continuity means "no surprises" — the function value at a point is exactly what you'd predict from nearby values.

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