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A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. "The plane" means the Euclidean plane.)

This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the question of how to define "rule." Defining it as a rule is also inconsistent with the standard definition of a function, since there can only be countably many rules.

It also avoids all the possible student confusion associated with heavy use of set-theoretic ideas and notation.

It isn't as general as the mathematician's general definition of a function, but that's a side issue, and the generalization can easily be carried out later.

A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. "The plane" means the Euclidean plane.)

This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the question of how to define "rule." Defining it as a rule is also inconsistent with the standard definition of a function, since there can only be countably many rules.

It also avoids all the confusion associated with heavy use of set-theoretic ideas and notation.

It isn't as general as the mathematician's general definition of a function, but that's a side issue, and the generalization can easily be carried out later.

A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. "The plane" means the Euclidean plane.)

This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the question of how to define "rule." Defining it as a rule is also inconsistent with the standard definition of a function, since there can only be countably many rules.

It also avoids all the possible student confusion associated with heavy use of set-theoretic ideas and notation.

It isn't as general as the mathematician's general definition of a function, but that's a side issue, and the generalization can easily be carried out later.

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source | link

A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. "The plane" means the Euclidean plane.)

This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the question of how to define "rule." Defining it as a rule is also inconsistent with the standard definition of a function, since there can only be countably many rules.

It also avoids all the confusion associated with heavy use of set-theoretic ideas and notation.

It isn't as general as the mathematician's general definition of a function, but that's a side issue, and the generalization can easily be carried out later.