Brownian space-time functions of zero quadratic variation depend only on time

Let B-t, t greater than or equal to 0, be a 1-dimensional Brownian motion a nd let f : R X [0, infinity[ --> R be a continuous function. We show that i f t bar right arrow f(B-t, t) is locally of zero quadratic variation, then f(x, t) = f(0, t) for all (x, t) is an element of R X [0, infinity[. This r esult extends recent work of F. B. Knight, thereby confirming a conjecture of T. Salisbury.