Lesson 21.4

Continuity and One-Sided Limits

Establishing the criteria for a function to be continuous at a point.

Introduction

Establishing the criteria for a function to be continuous at a point.

Past Knowledge

You understand limits and one-sided limit notation.

Today's Goal

We formalize continuity and identify types of discontinuities.

Future Success

Continuity is required for many calculus theorems (IVT, MVT, FTC).

Key Concepts

Three Conditions for Continuity at x = c

1. is defined
2. exists
3.

Types of Discontinuities

Removable (hole): Limit exists but ≠ f(c)
Jump: Left and right limits exist but differ
Infinite: Limit is ±∞ (vertical asymptote)

Continuous Function Types

Polynomials, exponentials, sine, cosine are continuous everywhere. Rational, log, tan have restricted domains.

Worked Examples

Example 1: Check Continuity (Basic)

Is continuous at x = 2?

1. f(2) = 5 ✓ defined

2. ✓ exists

3. Limit = f(2) ✓

Continuous at x = 2

Example 2: Removable Discontinuity (Intermediate)

Analyze continuity: at x = 1

1. f(1) undefined (0/0) ✗

2. ✓

Removable discontinuity (hole at x=1)

Example 3: Jump Discontinuity (Advanced)

For , check continuity at x = 2

1. f(2) = 5 ✓

2. Left:

Right:

3 ≠ 5, so limit DNE ✗

Jump discontinuity at x = 2

Common Pitfalls

Checking only the limit

All THREE conditions must hold for continuity.

Assuming piecewise = discontinuous

Piecewise functions CAN be continuous if pieces connect properly.

Real-World Application

Intermediate Value Theorem

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists c in (a,b) where f(c) = N. This guarantees solutions exist—used for finding roots graphically!