Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
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Sequences and Infinite Series Power Series Vectors and Geometry of Space Vector-Valued Functions
Calculus IV
15.Functions of Several Variables
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S = contains supplemental resources
Course: Calculus I
Topic: Limits and Continuity
Subtopic: Continuity

Overview

Whether or not a function's graph has holes, jumps, or asymptotes is consequential to evaluating limits and performing other operations in calculus. These features of a graph are examples of discontinuities. The concept of continuous functions is the focus of this lesson. A function is continuous over its domain if and only if its graph can be drawn without removing your pencil from the paper.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 1.10.1 Understand the definitions of continuity at a point, on an open interval, and on a closed interval
• 1.10.2 Use algebra, graphs, and/or calculus to find points of discontinuities
• 1.10.3 Distinguish between continuous from the left, continuous from the right, and continuous
• 1.10.4 Distinguish between removable and non-removable discontinuities
• 1.10.5 Rewrite a function that has a removable discontinuity as a piecewise function that is continuous over its domain
• 1.10.6 Know which common functions are continuous, which are not, and for those that are not, the value(s) at which they are discontinuous
• 1.10.7 Understand and be able to apply properties of continuity
• 1.10.8 Understand and be able to apply the Intermediate Value Theorem

Terminology

Define: continuity, discontinuity, continuous at a point, continuous from the left/right, continuous on an interval, continuous everywhere, removable discontinuity, non-removable discontinuity, intermediate value theorem (IVT)

Text Notes

This a vital section full of important terminology and theorems. Study it thoroughly! To learn the theorems, it helps to physically write or type them in your own words.

Supplementary Resources (optional)