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Consider general function $f:A\rightarrow\mathbb{R}^{n}$. Show that the graph of f in the plane is exactly the parameterized curve $F(t)=(t,f(t))$

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    Strictly speaking, the _graph_ of $f$ is the _image_ of $F$. A graph is a set of points in the plane, while a parametrized curve is a mapping into the plane. (Mathematicians tend to be a little careless with terminology, but it's important to keep these fussy details as straight as possible in your mind, especially if you intend to pursue graduate work.) – 2017-01-10

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If $f$ is discontinuous, it is difficult to say about the curve. For example, the graph of a discontinuous additive function $a:\mathbb{R}\to\mathbb{R}$ is dense on the whole plane. What kind of a curve is the graph of such a function?

Nevertheless we always have

$$ \text{Graph}(f)=\bigl\{\bigl(t,f(t)\bigr)\colon t\in A\bigr\}. $$