Surjection/Examples/Real Sine Function to Image

Example of Surjection

Let $I$ denote the closed real interval $\closedint {-1} 1$.

Let $f: \R \to I$ be the mapping defined on the set of real numbers as:

$\forall x \in \R: \map f x = \sin x$

where $\sin$ denotes the sine function.

Then $f$ is a surjection, but not an injection.

Sources

• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions: Example $1$