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⊆RI⊆R is an open interval, f:I→Rf:I→R is a function, and r∈Ir∈I is p-random, then f(r)f(r) is p-random provided1.f is p-computable on the dyadic rational points in I, and2.f varies sufficiently at r  , i.e., there exists a real constant C>0C>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.