For a function , its graph of a function is the subset of the Cartesian product . Exactly one ordered pair has first coordinate for each . For example, the quadratic function on the real line has the parabola as its graph.
The graph determines the values and the domain of a function, but alone does not record a larger specified codomain: the same set of ordered pairs can be regarded as lying in or when both contain all its outputs. It also differs from a graph in graph theory, whose edges need not specify exactly one output per input.
If is a compact space and is a continuous map, the graph of a function of is compact in the product topology on . Indeed,is a continuous map onto that graph, so the result follows from the continuous image of a compact space theorem. No Hausdorff assumption on is needed.
The coordinate projection map restricted to the graph is a continuous inverse to . Thus the graph is actually homeomorphic to . For instance, the graph of over has the same intrinsic topology as the interval, even though it is curved in the plane.
For a continuous map with Hausdorff space , its graph of a function is closed in the product topology. If , choose disjoint neighborhoods of those two points in . Continuity supplies a neighborhood of whose image lies in the neighborhood of ; its product with the other neighborhood misses the graph.
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The graph of a function is a visual representation of the relationship between the inputs (independent variables) and outputs (dependent variables) of that function. In coordinate geometry, a function can often be represented in a two-dimensional space using a Cartesian coordinate system, where the x-axis represents the independent variable (often denoted as \( x \)) and the y-axis represents the dependent variable (often denoted as \( f(x) \) or \( y \)).