I am reading an article "Differential Equations: Not just a bag of tricks" in the mathematics magazine. The author has given elementary examples of symmetry ($y=x^2$ symmetric about $y$ axis, $y=x^3$ symmetric about origin, $y=\sin x$ symmetric in translation by $2\pi$) and then proceeds to define:
These transformations are symmetries of $f$ because they map the graph of $f$ to itself. In general, for a function $f : \mathbb{R} \rightarrow \mathbb{R}$ , a symmetry of $f$ is a continuous map from $\mathbb{R}^2$ to $\mathbb{R}^2$ that maps the graph of $f$ to itself and has a continuous inverse.
I do not understand why the introduction of $\mathbb{R}^2$ was needed and why the inverse was mentioned. Moreover, what is the importance of stressing continuous. Thank you.
The graph of a function from $X$ to $Y$ is by definition the set of pairs $(x, f(x))$, where $x \in X$ and this is a subset of $X \times Y$ (for a set theorist, the function is the graph). Hence for real-valued functions on $\mathbb{R}$ we get that the graph is a subset of the plane. The continuity is, I suppose, to avoid trivialities: all graphs can be bijectively mapped onto itself by a function of the plane (the all have the same cardinality) and we can extend these maps to bijections of the plane, set-theoretically. So we need to restrict the class of maps that can be considered, or all there would be too many symmetries.