You are describing a tankless water heater. One of the main purposes of having a tankless water heater is to save money. Tank type water heaters require heating and continueheating a body of water prior to use. A tankless water heater only heats the water once which is also known as "On demand".
Other benefits of a tankless water heater are 1. They take less space. 2. They (providing they are sized correctly) provide an endless supply of hot water. Tank type water heaters run out of hot water when the hot water in the tank is replaced by colder incoming water. Even though this is a benefit, it is a double edged sword. You can stand in a shower all day and have still have hot water but you will pay a bigger energy bill.
Tankless water heaters cost more initially but are supposed to save more in the long run. Tankless heaters are becoming more popular because of their energy saving benefits.
Hope this helps.
