Proving that winding number is constant in a set created by the partitioning of plane into sets by a curve In page-339 of Tristan Needham's Visual Complex Analysis, an argument that the winding number in any one of the set which curve partitions plane into must be constant. It is done by considering the $\nu(L,p)$ function which takes in a curve $L$ and a point $p$ and tells the number of windings of $L$ about $p$.
The proof is as follows:

Since the winding number of $L$ is just the sum of rotations due to all its segments, it follows that it too depends continuously on location of $p$ unless $p$ crosses $L$. In other words, if we move $p$ by a little bit then the rotation angle will change by a tiny bit. A tiny movement from $p \to \tilde{p}$ can only produce a change of $\left[ \nu(L,\tilde{p}) - \nu(L,p)\right]$ in the winding number but since this small difference is an integer it is exactly zero.

The above makes some sense to me but I can't understand the point preceding it,

As z traverses a short segment of L, the rotation of $z-p$ will continuously depend on $p$ unless $p$ crosses $L$

And it is given as a footnote :

Consider behaviour of the rotation a short segment of L as p crosses it

I can't understand what the author is trying to say here, could someone lend a more direct explanation?
 A: Suppose we have a short piece of a (piecewise differentiable) curve $L$, and a fixed point $p$ (not on $L$, say). Then as $z$ moves along the curve, the angle/argument of the vector/number $z-p$ will change by some small (possibly negative) amount $\theta$, contributing $\theta/(2\pi)$ to the winding number of the curve $L$ about $p$.
Now, that $\theta$ or $\theta/(2\pi)$ depends on the choice of point $p$. If you move $p$ a little bit but don't cross $L$, the traversed angle $\theta$ won't change by very much. That is what is meant by

As $z$ traverses a short segment of $L$, the rotation of $z−p$ will continuously depend on $p$ unless $p$ crosses $L$

The footnote is asking you to imagine the case of $p$ crossing $L$ as well. The tiny angle $\theta$ will change dramatically then, but that doesn't matter for the main "winding number is the same in each region" idea.
Edit: A typical example of what happens when $p$ moves slightly to cross a short piece of $L$ is shown in this hasty sketch:

If the short piece of $L$ goes upwards, and $p_1$ is just to the left of it, then $\theta_1$ is just under $\pi$ since $z-p_1$ goes from just-to-the-right-of-downwards to just-to-the-right-of-upwards. But if $p_2$ is just to the right of it, then the orientation has swapped, making the rotation clockwise and the angle negative, so that $\theta_2$ is some value just above $-\pi$ instead.
Edit 2: I'll explain how this affects the winding number. If you move from $p_2$ to $p_1$, then you've changed from a local angle of $\theta_2\approx-\pi$ to an angle of $\theta_1\approx\pi$, which means you've added $\theta_1-\theta_2\approx2\pi$, or $1$ revolution, bumping up the winding number assuming the changes from the other parts of $L$ aren't significant. A slightly different diagram and argument for this is given on the very next page of Needham, p. 340.
