Definition:Graph of Real Function

Definition

Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.

Let $f : U \to \R^k$ be a real function.

The graph $\map \Gamma f$ of the function $f$ is the subset of $\R^n \times \R^k$ such that:

$\map \Gamma f = \set {\tuple {x, y} \in \R^n \times \R^k: x \in U \subseteq \R^n : \map f x = y}$

where $\times$ denotes the Cartesian product.

Sources

• 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1.1$: Smooth Manifolds. Topological Manifolds
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics