This is, indeed, how electric stoves (flat top, coil top, or otherwise) work. It's also how most microwaves work, incidentally. The heating elements have two states: on or off. If you set a burner to an intermediate setting, the way it achieves that intermediate temperature is by turning on for a particular amount of time, then turning off for a particular amount of time.
Note that the burner doesn't actually cool down all that quickly just because it momentarily doesn't appear red. This is why it's so easy to burn things on an electric cooktop if you're not used to it: burners stay hot enough to cook food for quite a long time after being turned off. If you're done cooking something, you need to physically move it off the stove, or at least off of any burner that has been turned on in the last, oh, half an hour, otherwise you will continue cooking it, thus potentially burning it.
To test whether a burner is working, turn it to the highest heat and check that it stays on more or less constantly. Note that depending on your stove, you might have to put a pot on the burner for it to turn on at all. And if your stove comes equipped with a "boil-over" sensor, all bets are off: the burner may not stay on constantly no matter what you do. In any case, the failure state of most stoves is "stops working at all", not some sort of "works when it feels like it", so if you can get your stove to produce heat, chances are it's working as correctly as it's capable of.