# Increse/decrease in day light hours at locations very close to the Arctic and Antarctic circles

I am not an expert in geography. In fact, I know almost nothing about geography. This question in fact concerns a math problem: I have been assigned to construct an analytic description, i.e. to find a functions which takes in coordinates on the earth as input, and splits out the daylight hours at that position, during the time of winter solstice.

Right now I have been successful in finding the correct function. The problem comes with interpretation. With my solution I come to the conclusion that, when the position is very close to the Arctic and Antarctic circles ( but have yet reached within ), my solution says that I should expect some very rapid, waterfall like drop in the day light hours. Mathematically speaking this means that the derivative of my function is infinite at the boundary of the Arctic and Antarctic circles.

I find this hard to believe. If this is true, then based on my observations from the map, would say things like, a small island on the boundary of the Arctic circle may still have some good day light, but once you enter the circles you get no day light at all.

Of course the small should be relatively different. In my calculation I just assumed the earth to be a unit ball with radius one, which certainly isn't the case.

So my question is, is this actually true, or possibly true in any sense?