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

Let $B_ t$, $t\ge 0$, be a $1$-dimensional Brownian motion and let $f\colon {\mathbb{R}}\times [0,\infty [\,\to {\mathbb{R}}$ be a continuous function. We show that if $t\mapsto f(B_ t,t)$ is locally of zero quadratic variation, then $f(x,t)=f(0,t)$ for all $(x,t)\in {\mathbb{R}}\times [0,\infty [$. This result extends recent work of F. B. Knight, thereby confirming a conjecture of T.Salisbury.