[Geezer]

It's more like 50% per day with my water heater.

Good grief! Are you heating a cauldron on a wood stove?

[jo nurd]

This is all good discussion and the theory is correct but one thing is missing.  In many countries where hot water is heated with electricity the electrical power to the heater is controlled remotely by the power company.  There are two tariffs for the cost of power – one or power consumed during the ‘peak’ hours and one for ‘off peak’.  The reasons for this are another topic. Peak is usually a few hours in the morning, lunch time and the evening.  So . . . if the hot water heater is switched on at 2 am it will heat up but if it is switched on at 6am it will not heat up until the morning peak is over (albeit some installations have an override switch BUT the override uses electricity at the peak tariff).  So . . . to save energy 1) take a shorter shower and 2) use a solar assisted or heat pump hot water supply or a well insulated hot water heater with the smallest capacity your household can manage with off peak demand heating.  Clearly option 1 is the only option for the person with the original question.

[syhprum]

If my water came out of the hot tap at 46°F I would regard that as perishing cold not blistering hot.

[CliffordK]

If my water came out of the hot tap at 46°F I would regard that as perishing cold not blistering hot.

I wonder if that was intended as Celsius.

That is about the temperature of the Metolius River. 
I will tell you that it is CCCCOOOOLLLLDDDD

I've been boating in it.  Get splashed with it and it bites.
I did tip over once, but I didn't stay in the water for very long.  [xx(]

I believe well water around here comes out in the mid-50's.

[Geezer]

If my water came out of the hot tap at 46°F I would regard that as perishing cold not blistering hot.

Sheesh! You scientists are soooooo literal. My comment was an example of irony.

Wiki - Irony "Ironic statements (verbal irony) typically imply a meaning in opposition to their literal meaning."

[JK]

Most responses here have missed the point. The point is NOT whether heating in advance is economical, but whether switching on and off is economical. Lets assume there is some heat loss over a period of time. That would mean at some point the water would need to be reheated. Let's say I lived in Africa, where electricity and water were scarce and I only showered once a month. Surely it would be less expensive to turn on the geezer just before I had a shower, and as soon as the water was hot, I'd use it. Then leave it switched off for the rest of the month. Whereas keeping it on for the entire month, it would consume more electricity to keep the geezer on, so that any heat lost could be regained.

And do the geezers remain on, at a low temperature? Or do they turn off completely when the water is hot enough? If the later is the case, then turning it off yourself is the same thing. Except it would never turn itself on again when the temperature dropped, so you'd being saving power.

BTW, instant gas hot water systems have been around for decades. My Dad is using one that's at least 50 years old and still working perfectly well.