Lighting, Light Fixtures, Ceiling and Exhaust Fans - Ceiling Fan Switch Question

I had to replace a switch but whwn I went to the store I found 'Single continuity function' and 'Split continuity function' switches. What's the difference?

Is this the internal fan switch or a wall switch? Never heard that term for switches.

I have heard the term but honestly not sure what it means. If i have "continuity" it means there are no breaks in a wire or circuit between 2 points. I would just replace the switch that was there before with another like it.

It's an internal, 3 speed and off type switch, and both types of switches look alike at Home Depot. Not sure that asking an associate would give me an accurate answer, so I'm relying on you 'pros'.

A: What color is the shell ( red, black, other)?

B: what capacitor is in there ( how many wires)?

White

Don't recall.

Here's the whole story. The original switch (white body) broke off at the threaded neck. I purchased a replacement white bodied switch at Home Depot, installed it, but it didn't work. So I glued the original one together and reinstalled it. It works now but I'm sure it will break again in the future.
The switch that I bought was a 'single continuity' type aand had a white body. The 'split continuity' ones had a black body.
The fan has been reassembled so I can't see what kind of capacitor is in it, nor how many wires it has.
The website for the manufacturer doesn't explain the difference, but I'm still interested in knowing the difference.

What markings are on the new and old switch?

I did a little searching.

Apparently, single continuity sequentially connects each output terminal to power, each speed, and is used with a 3-tapped winding and a sigle tap capacitor.

Split continuity uses a dual tapped capacitor and untapped winding, or a single tapped capacitor and double tapped winding. It connect one winding/cap, the other, or both to achieve the electrical properties of the different speeds.