Definition:Square/Function/Real

Example of Real Function

The (real) square function is the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$

Graph

A graph of the square function on $\R$ can be presented as:

Properties

Real Square Function is not Injective

Let $f: \R \to \R$ be the real square function:

$\forall x \in \R: \map f x = x^2$

Then $f$ is not an injection.

Real Square Function is not Surjective

Let $f: \R \to \R$ be the real square function:

$\forall x \in \R: \map f x = x^2$

Then $f$ is not a surjection.

Also see

• Domain of Real Square Function
• Image of Real Square Function

• Definition:Integer Square Function

• Results about the square function can be found here.

Sources

• 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.2$: Functions
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $8$: The System of the World: Calculus