MATH119 - Functions, Limits, and Continuity

This is the first in a series of courses that cover the equivalent of a one-year college course in calculus. These courses are

• MATH119 - Functions, Limits, and Continuity (This course)
• MATH120 - Differential Calculus for Measurement
• MATH121 - Sequences, Limits, and Series
• MATH122 - Integral Calculus for Measurement
• MATH123 - Series, Approximation, and Sensitivity

This course introduces the concepts of function, limits, and continuity. These concepts, and the theorems that are presented on these concepts, are the building blocks for defining the concepts in differential and integral calculus of one variable.

This course has three major sections, Functions, Limits, and Continuity. The first section introduces some notation that will be used in this series of courses, but other than that should be a review of concepts that should have been learned in Algebra. The concept of limit is the fundamental building block that calculus is built upon, which is the subject of the next section. The final section covers the topic of continuity.

The course outline is as follows:

• Introduction
• Functions
  • Definition
  • Notation
  • Graphs
  • The Roots of a Function
  • The Inverse of a Function
  • Summation Notation
  • Product Notation
  • Examples
    • Linear functions
    • Polynomials
    • Rational Functions
    • Algebraic Functions
    • Even and Odd Functions
    • Periodic Functions
    • Trigonometric Functions
    • Exponential and Logarithmic Functions
    • Hyperbolic Functions
• Limits
  • Definition of limit
  • Examples of Limits
  • Left and Right Sided limits
  • Infinite Limits
  • Limits at Infinity
  • Theorems Involving Limits
• Continuity
  • Definition of Continuity
  • Examples of Continuous and Discontinuous Functions
    • The Unit Step Function
    • The Tangent Function
    • A function continuous nowhere
  • Removable Discontinuities
  • Essential Discontinuities
  • Step Discontinuities
  • Piecewise Continuous Functions
  • Theorems Involving Limits and Continuity
• Summary

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