Let a > 1. Assume that ap+q = apaq and (ap)q = apq for all p,q ˆŠ Q, and that ap For each x ˆŠ R, define

a) Prove that A(x) exists and is finite for all x ˆŠ R, and that A(p) = ap for all p ˆŠ Q. Thus ax:= A(x) extends the "power of a" function from Q to R.
b) If x, y ˆŠ R with x c) Use Example 2.21 to prove that the function ax is continuous on R.
d) Prove that ax+y = axay, (ax)y = axy, and a-x = 1/ax for all x. y ˆŠ R.
e) For 0

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A(x) := sup{a": q e Q and q x).