I'm buying a new set of lights and I'm curious if there are any lights out there that are both flash and continuous lights. A set with a switch which allows the user to choose between these two options would be ideal.
Yes. For example, the Canon Speedlite 320EX and Speedlite 270EX II.
This is for video applications. For simply seeing where light will fall, many flashes have a "Modeling light" function, which rapidly strobes the flash for a few seconds so you can see where the shadows are.
Most mains-powered studio flash would easily qualify. Only AC-powered lights at the "toy" end of the spectrum and a handful of highly-specialised fixtures lack modelling lights. The modelling lights are not always bright enough for many types of photography or videography, but the more powerful flashes tend to have 250W halogen bulbs, which are entirely adequate for video or still life (and even for wide-aperture portraiture at moderate ISO settings). Any modifiers that work for the flash (softboxes, reflectors, grids, spots, fiber optic light pipes, etc.) will do double-duty as well. This is particularly the case for lights using a frosted dome over the modelling lamp and flash tube (Paul C. Buff Einsteins, Profoto and Elinchrom's Style range among them) which minimizes the difference between the modelling light's apparent position and the flash tube's. Portable flash usually has only vestigial modelling light, if any at all, since using a modelling light will significantly reduce the number of flashes available per battery charge.
Some LED panels, which are aimed primarily at the video market, also offer a "flash mode" in which the LEDs are driven to a power level they can't sustain normally (they'd burn out) for a very brief period. It's not quite flash (the duration is normally longer than a flash, often long enough to enable something like high-speed sync, and they don't get quite as bright as a xenon arc), but it's a substantial boost over the normal LED output level.
The big studio lights have a model lights inside, usually a halogen spot inside the Xenon ring tube. The strobe I used for work has a maximum of 100W for the modelling light. IF you have a set of these, you are probably ok for shooting with it, but it is more for setting up the scene. This also makes the white balance different.
I assume you meant the same light source. For that you need to look to machine vision lights like Advanced Illumination.
Their lights can be chosen on all sorts of single wavelengths incl. NIR, or white and they have a very long lifespan and keep the brightness throughout their life, but have a small area of interest (good for macro work or product/food photography). You buy them with a control box that can be continuous and flash and you can set the brightness. You can also get them in many different shapes (ring, spot, darkfield/brightfield/both, diffuse, plates, coaxial w/ prism, lines, patterns). They are very expensive, typically $2000-$4000, though. But they use the same light source for continuous and strobing (flashing) light.