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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.

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The graph $G$ of $f$ is by definition a subset of the plane: $G=\{(x,y);y=f(x)\}$. –  Did Jun 8 '11 at 14:33
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Although most mathematics can be based on set-theory, this is unfitting the tag. I'm not sure that the [functions] tag can stand for itself, but I'm not sure what can be added instead. –  Asaf Karagila Jun 8 '11 at 14:40
    
@Asaf I'd like to know what field of mathematics discusses these things from the start. Perhaps that might settle the tags as well. –  kuch nahi Jun 8 '11 at 14:43
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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.

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$\mathbb{R}^2$ is here because we want to discuss transformations of the graph of the function, which lives in $\mathbb{R}^2$.

The inverse and continuous were required in order to limit the class of symmetries to only "natural" ones. Without them, you can have "symmetries" that take the points of the graph, do a complete mix-up of them, and place them again on the graph at a completely random fashion. This is not usually the sort of transformations we are interested in; we want some sort of structure of the geometry of the space (in our case, $\mathbb{R}^2$) to be preserved.

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Thanks. What field of mathematics is this where these kinds of things are discussed on a starter level. (Also, permutations on a discrete set can have no symmetry?) –  kuch nahi Jun 8 '11 at 14:38
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For discrete sets a different definition of symmetry will be used; context is everything. I'm afraid I can't recommend books that discuss these sorts of things specifically. –  Gadi A Jun 8 '11 at 14:40
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