Lesson 21.4

Continuity and One-Sided Limits

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

Introduction

A function is continuous if you can draw its graph without lifting your pencil. Formally, continuity at a point requires three conditions: the function is defined, the limit exists, and they're equal.

Prerequisite Connection

You understand limits and one-sided limit notation.

Today's Increment

We formalize continuity and identify types of discontinuities.

Why This Matters

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!