Function vs. map
“Function” and “map” (or “mapping”) mean exactly the same thing. “Map” is more common when the function preserves some structure or acts on geometric objects, but every statement about functions holds for maps, and vice versa.
Definitions
Level:
Missing definition: function
Missing definition: map
Examples
| Example | function | map | Why |
|---|---|---|---|
| $f(x) = x^2$ on $\reals$ | ✓ | ✓ | The squaring function, as a function $f : \reals \to \reals$. |
| A linear map $T : \reals^2 \to \reals^2$ | ✓ | ✓ | |
| A continuous map from a sphere to the plane | ✓ | ✓ |
Notes
- Some authors reserve “function” for maps whose outputs are numbers (real or complex), and use “map” for everything else. This is a convention of style, not a different definition.
- In category theory, “morphism” generalizes “map” to arrows that need not be functions at all.