what does R^R mean where where the R's are bold?

[alamps3]

what does R^R mean where where the R's are bold? Specifically I'm looking at the Princeton Review math subject test book, example 6.24 where it asks, "Is the ring R^R an integral domain?"

[lime]

All continuous functions R->R.

Check couple of pages before. Right before Example 6.19.

[cowingzitron]

The answer you received from username 'Lime' is not correct. The notation in analysis for the set of all continuous functions from R to R is C^0(R). The zero indicates that the functions are continuous but not assumed to have derivatives everywhere; C^2(R) would be twice continuously differentiable functions, i.e. f''(x) exists and is continuous everywhere.

The notation you saw, R^R, with an exponent R, is the set on ALL functions from R to R, regardless of continuity. It is a purely set-theoretic notion, analogous the notation 2^X for some set X; that indicates the powerset of X, since any choice of a subset of X involves assigning to each point in X the value 0 or 1 based on whether the point is in the subset.

The distinction between C^0(R) and R^R is important, since there are certainly non-continuous functions from R to R, e.g. the dirac delta function. Not only does the distinction matter from the point of view of analysis, it also matters for set theory, since R^R has strictly greater cardinality (beth 2) than C^0(R) (beth 1).

[lunarmono]

The dirac delta function is not a function from R to R...