I believe that continuous lighting can give just as good results. A couple of minor drawbacks, though..... unless you get something towards the 'top of the range' you probably won't be able to control the output of a light without having to turn it away or move it back (though some have 'barn doors' so you can concentrate the light). It's usual to have a lighting setup that is stronger on one side than the other - with flash it's done by reducing the power output on one lamp, but with cheaper lamps it has to be done by moving the lamp away.
The big bonus of continuous lighting is that you see exactly what effects your setup is having when you move it around, without needing to take test shots as most people would with flash.
They used to be called 'hotlights' - because they literally were very hot to work with - but the use of daylight long-life bulbs means they're cooler than other kinds of portable lamp. These work at a light temperature more like daylight - the older ones were tungsten so you had to make adjustments for it with your White Balance or with processing if using Raw (or, in pre-digital days, with a different film). If you're happy with White Balance corrections, in camera or in post processing, there's nothing wrong with getting tungsten lights, which you might get 'second hand' and should be a bargain. You may even be able to change the bulbs over to Daylight but most of the older lamps use a different type of bulb altogether, with a different fit.
You can get new, daylight continuous lighting lamps and stands a lot cheaper than flash units but you need to check that they have a slot that can carry a reflective umbrella. The ideal 'continuous lighting' lamp would have at least two power settings, barn doors, umbrella reflector holder and a diffusion method, preferably a softbox. If you're just starting out, though, cheap and cheerful is good enough to learn on.
Worth checking out
this video - otherwise, look out for Cameron Lighting Kits.