Continuity — No Breaks, No Jumps

A function is continuous at x = a if three conditions are met:

1. f(a) is defined

2. lim(x→a) f(x) exists

3. lim(x→a) f(x) = f(a)

Types of discontinuity:

Removable (hole): Limit exists but f(a) is undefined or different. Can be "fixed."

Jump: Left and right limits exist but differ.

Infinite: Function goes to ±∞ (vertical asymptote).

Intermediate Value Theorem (IVT):

If f is continuous on [a,b] and k is between f(a) and f(b), then there exists some c in (a,b) where f(c) = k.

Translation: a continuous function must hit every value between f(a) and f(b). No skipping.

AP exam loves IVT. Expect a free response question requiring you to justify using IVT.

Reference:

Wikipedia: Continuity

https://en.wikipedia.org/wiki/Continuous_function