Functions That Preserve p-Randomness

We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general theorem: If I ⊆ R is an open interval, f : I → R is a function, and r ∈ I is p-random, then f(r) is p-random provided 1. f is p-computable on the dyadic rational points in I, and 2. f varies sufficiently at r, i.e., there exists a real constant C > 0 such that either (∀x ∈ I − {r}) [ f(x)− f(r) x− r ≥ C ] or (∀x ∈ I − {r}) [ f(x)− f(r) x− r ≤ −C ] . Our theorem implies in particular that any analytic function about a p-computable point whose power series has uniformly p-computable coefficients preserves p-randomness in its open interval of absolute convergence. Such functions include all the familiar functions from first-year calculus.