I'd say the graph's qualitative behavior is correct. Once you get to the arctic circle, there are singularities in the rate of change of the length of daylight.

For example, if you're at the North Pole, the sun is below the horizon all winter, and then on the vernal equinox rises above the horizon and does not set again until the autumnal equinox. So, formally, the rate of change in daylight is zero all year long, except on the equinoxes when it is infinite. Any latitude above the Arctic Circle will have these kinds of singularities.

In practice there are some corrections to the amount of daylight that I discuss at the bottom of the article, the most important of which is the effect of atmospheric refraction. If you were standing on the North Pole, you'd actually observe the Sun appear to rise some time before the vernal equinox.

It's a shame we can't ask someone at the North Pole right now, a few days before the equinox, if the sun is up.

For what it's worth, brr.fyi documents a sunset at the South Pole: (https://brr.fyi/posts/sunset) and a sunrise (https://brr.fyi/posts/redeployment-part-one)

The graph looks like it's straight up interpolating between a sine curve and a tangent curve, it's so cool that way. (The calculations are more involved than that).

Also signal analysis people will enjoy this natural system producing "almost pure" triangle and square waves. Nudge the plot to 66.55 degrees and it's at the most triangle wave point. :)

I guess the graph doesn't show the location of the sun even with daylight.

In the extreme latitudes, you have short days where the sun just goes maybe 5 degrees into the sky and then down again: https://www.youtube.com/watch?v=Wzv_3LeA85c

And days where the sun just circles around your location on the planet: https://www.youtube.com/watch?v=in0B1OQG3-M

The opposite also happens, where the sun is below the horizon for months or weeks, and at the end of that period, it will inch towards the horizon, and you expect sunrise, but nope, it'll move furrther again... of course there'll be twilight (followed by night), but you might not see the orange ball of bright light for weeks or months.

Yes. At these latitudes there are times of the year when it's always night or always day. This messes with the math such that the graphs look weird.