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Given a real-valued function$f\colon \mathbf{R} \to \mathbf{R}$, thegraph of$f$ is the set of all input-output pairs$(x,f(x))$ regarded as a set of points in the plane$\mathbf{R} \times \mathbf{R}$. Considering the graph of a function gives us a geometric perspective on the data that the function represents.

• If the function$f$ iscontinuous, the graph of$f$ "looks continuous." That is, there are no gaps, and the graph is aconnectedcurve.

• If the function$f$ is differentiable, then it will contain no "sharp corners."

• If we're thinking of the domain of the function as representing time, the the graph gives us a nice visualization of the change in outputs of the function over time.

A graph can be defined much more generally though. Let$\mathbf{k}$ be a local field, and suppose$f$ is a vector-valued function$f\colon \mathbf{k}^n \to \mathbf{k}^m$ where$f(x_1, \dotsc, x_n) = (y_1, \dotsc, y_m)$ and each coordinate$y_i$ of the output is a function of the$x_1, \dotsc, x_n$. In this setting, the graph of$f$ is the set of points

$$(x_1, \dotsc, x_n, y_1, \dotsc, y_m) \subset \mathbf{k}^{n+m}\,.$$

This general construction of the graph of a function can be useful in the study ofalgebraic geometry or the study ofmanifolds.