• A link to early use of mathematical symbols. The symbol f(x) to denote a function was apparently first used by Euler in 1734.

• Notation and the concept of functions and graphs in The History of Calculus

• Definition of a function in terms of sets (domain and range)

• Graphing functions.

• Inverse functions and their relationship to the graph of f(x).

• Implicit Function Theorem , used for solving equations of the form f(x₁,x₂,x₃...x_N)=0

• Open sets and their role in understanding behavior of functions

Properties of functions

• Continuity of functions

• Differentiability

• Functions of a single variable

• Functions of more than one variables

• Functions of a complex variable

• Extensions, closure

Early questions (problems)

• Is every function continuous except at a countable number of points?
• Is every function continuous except for a set of measure zero?
• Is every function differentiable except for a set of measure zero?
• Is the limit of a sequence of continuous functions continuous?