Determine whether a point lies inside the curve or outside a random curve using pencil and scale

What the OP really wants, I think, is a crossing number algorithm for a simple closed continuous curve. Such a curve can be considered, however, as the limiting case of a simple polygon, for its edges becoming infinitesimally small. A relevant reference therefore may be this Wikipedia one:

• Point in polygon

and especially the crossing number algorithm.
But there is much more to tell about the Inside / Outside Problem, once you take a good look at it.
After some (in vain) attempts to answer the question in a concise manner, and while making too many duplicates of existing material, I've decided to simply redirect to my best shot so far:

• Inside / Outside Problem

If the points of the simple closed curve are considered as pixels (i.e. integer coordinates), then we can even devise a never fail algorithm (only assuming that our picture is not too large).

It is seen in the picture below that, for a continuous curve, simple and straightforward application of the crossing number algorithm will be OK for the points $A$ and $B$, but it certainly will go wrong for the points $C$ and $D$. According to Murphy's law , Anything that can go wrong will go wrong. Meaning that ignoring special cases, especially in a computational geometry environment, sooner or later, will be disastrous. It is noticed that the "wrong" cases have to do with rays that are tangent to the curve.