Calculus - Continuity

Continuity

Definition

f is continuous at a if:

1. f(a) exists
2. $\lim_{x \to a} f(x)$ exists
3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuity

• Removable: Limit exists but doesn’t equal function value
• Jump: Left and right limits exist but differ
• Infinite: Limit is ±∞

Properties of Continuous Functions

• Sums, products, quotients of continuous functions are continuous
• Composition of continuous functions is continuous
• Polynomials are continuous everywhere
• Rational functions are continuous on their domain

Intermediate Value Theorem

If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c ∈ (a,b) such that f(c) = k.

Extreme Value Theorem

If f is continuous on [a,b], then f attains both absolute maximum and minimum values on [a,b].