Mastery of Continuity: Curriculum Overview

This curriculum provides a structured path to understanding the mathematical concept of continuity, moving from the intuitive "pencil-on-paper" definition to the rigorous three-part limit definition. You will explore how functions behave at specific points and over entire intervals, while learning to classify various types of mathematical "breaks."

Prerequisites

Before beginning this module, students should have a firm grasp of the following concepts:

• Functions & Domains: Identifying where a function is defined, particularly for rational and radical functions.
• Intuitive Limits: Understanding what it means for a function to approach a value $L$ as $x$ approaches $a.
• One-Sided Limits: The ability to calculate and distinguish between \lim_{x \to a^-} f(x)$and$\lim_{x \to a^+} f(x)$.
• Basic Algebra: Manipulating rational expressions to find holes or vertical asymptotes.

Module Breakdown

Learning Objectives per Module

Module 101: Continuity at a Point

• Explain the three formal conditions required for a function f(x) to be continuous at point a$:
  1. $f(a)$ is defined.
  2. $\lim_{x \to a} f(x)$ exists.
  3. $\lim_{x \to a} f(x) = f(a)$.

Module 102: Types of Discontinuities

• Removable Discontinuity: A "hole" in the graph where the limit exists but doesn't match the function value.
• Jump Discontinuity: Where the left-hand and right-hand limits exist but are not equal.
• Infinite Discontinuity: Where the function approaches $\pm\infty (vertical asymptote).

[!TIP] A removable discontinuity can be "fixed" by redefining a single point, whereas jump and infinite discontinuities are inherent to the function's structure.

Module 201: Continuity over an Interval

• Define continuity on an open interval (a, b) as being continuous at every point within.
• Apply the definitions of Left-Continuity and Right-Continuity to satisfy requirements for closed intervals [a, b].
  • Right-continuous at a: \lim_{x \to a^+} f(x) = f(a)
  • Left-continuous at b: \lim_{x \to b^-} f(x) = f(b)

Module 301: The Intermediate Value Theorem (IVT)

• Understand that if f$is continuous on$[a, b]$and$L is any value between f(a)$and$f(b), then there exists at least one c$in$(a, b)$such that$f(c) = L.

Success Metrics

To demonstrate mastery of this curriculum, the student must be able to:

1. Perform the Three-Step Check: Successfully verify continuity for a piecewise function at its boundary points.
2. Classify Failures: Given a graph or equation, correctly label a discontinuity as Removable, Jump, or Infinite.
3. Predict Domain Continuity: State the intervals of continuity for any polynomial or rational function without viewing the graph.
4. Existence Proofs: Use the IVT to prove that a polynomial has at least one root within a specific interval (e.g., proving x^3 - x - 1 = 0 has a root between 1 and 2).

Real-World Application

Engineering & Physics

Continuity is essential in modeling physical systems where sudden "jumps" are impossible. For example, the velocity of a vehicle must be a continuous function of time; a car cannot be going 20 mph and then instantly 60 mph without passing through every speed in between.

Computer Graphics

In 3D modeling and animation, splines and curves must be continuous to ensure smooth surfaces. If the functions defining a character's movement were discontinuous, the character would "teleport" or jitter across the screen during rendering.

Economics

Market equilibrium models assume continuous supply and demand curves to ensure that an equilibrium price P$ actually exists. The Intermediate Value Theorem is frequently used to prove the existence of these stable economic states.