Continuity of spherical averages on a closed radius interval

Let $u$ be a real-valued continuous function on the closed ball of radius $R\ge0$ about $x$ in $\mathbb R^n$. Its normalized spherical average $A(t)$, defined using the fixed unit sphere and the map $\omega\mapsto x+t\omega$, satisfies

This supplies the continuity at radius zero and at the outer radius needed to pass from a radial derivative identity to a mean-value formula. In dimension zero the defining spherical measure is zero, and the average remains the constant zero function.