differentiation - Physical meaning of divergence

While reading the section on Hamiltonian mechanics in Taylor's Classical mechanics, I realized that I didn't fully understand what he was saying when he was explaining why $$\nabla\cdot\vec{F(\vec{x_0})}=\frac{1}{V}\frac{dV}{dt}, $$for small volumes around $\vec{x_0}.$

$$\frac{dV}{dt}=\int_{\partial V}\vec{n}\cdot\vec{F}dA=\int_V\nabla\cdot\vec{F}dV.$$ Taking over a small enough volume, $\nabla\cdot\vec{F}$ is constant, factors out of the integral, and the result follows.

I'm not sure why $\nabla\cdot\vec{F}$ is locally constant. The first time I read this, I thought is was just a matter of assuming sufficient smoothness on the partial derivatives, but after thinking for a second, that can't imply locally constant. So I'm a little confused.

Answer

Think about it one more time. If $\vec{F}$ has continuous partial derivatives, then $$\vec\nabla\cdot\vec{F}=\sum_i \frac{\partial F_i}{\partial x_i}$$ is also continuous. If a function is continuous, it's approximately constant on sufficiently small volumes: that's pretty much the definition of continuity! So your original understanding was just fine.

Maybe your confusion is on what locally constant means? It doesn't mean that the function is actually constant on any given region, just that as the region gets smaller and smaller, the variation of the function over the region tends to zero.