Hot ..Cold ..Hot ..Cold

When I take a shower recently, the water will stay warm for maybe 3 minutes, then it turn very cool for 1 minute then another 3 minutes is hot, and keeps repeating this cycle. When the kitchen faucet is left running for a length of time the same cycle appears. I've checked the water heater elements and they both show 16 ohms, so no burnt out element. Any ideas??

PS The well water goes through an indoor heat pump to pre-heat it before it gets to the main electric water heater.

only thing i can think of is your cold water tube broke off...it directs all incoming water to bottom of heater tank.

Cold water tube would be my guess too.

Is your well pump cycling with the temperature changes?
If so, could be a bad expansion tank not holding pressure steady.

Your wife is messing with you.

Is it just in the shower?? It might be the anti-scald dohicky in the faucet. I've heard of them doing that once in a while, when they go bad.

It's called a dip tube but I think it would go to cold water after 3 minutes and stay cold
Dip Tube

That's called the "Old Lady Syndrom", hot..cold..hot..cold.......LOL.
In another life I worked at a Homeless and Desperate store in the plumbing department. I told many customers, "you need a new dip tube dip stick". Your description leads to that conclusion, especially since you said the kitchen faucet has the same pattern. The tube is under the inlet opening (where your cold water enters the water heater). The tube should be almost as long as the heater is deep. If you are able to pull it out and it's short that is your problem. Most are a piece of plastic tube with a flare on the top side to keep it from falling in the tank. In the old days it was just easier to replace the heater because they were pretty cheap, now days I'm not so sure since I've not purchased one in ages. HTH

The usual shots in the dark have already been offered,let's try something systematic for a change.
You didn't mention the volume or pressure changing so I assume they don't.
Turn the shower on then feel the pipe as it exits the heater. If the pipe alternates between hot and cold,that eleminates everything down stream including the shower hot&cold controls. I don't know exactly how the heat pump is plumbed in but it probably would be playing a part in this eventhough it's plumbed correct. If the dip tube is broken inside the heater"AND"the pump is cycling about ever 3 minutes,that could cause the osulation in tempature. Set the heat so it does not cycle the repeat the test above. If the water runs hot then cold and stays cold,90% odds are the dip tube is broken off. If there is no effect,move to the shower facuet.
Single handel controls are known to do crazy things. Rebuild or replace it. If you have dual knobs,there is a chance the problem can be solved by replacing both washers and/or cartridges. Still no joy??? Let's move to the laundry. Turn off either supply valve to washer and repeat test. If that stops the problem,replace the mixer valve on washin machine.
We will stop here until you have gone through all that and repost. Good luck.

I agree with Royce. If you have a well and the air pressure in your pressure tank is low, it will act this way... assuming you have a well with a pump and not on city water pressure. I had the same problem and was able to correct it by adding air to the air bladder in the pressure tank. For some reason the air had leaked out and was the cause of the constant cycling in pressure from the pump and thus influx of cold well water.