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 com-putable 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 ≥