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I know that a camera has a filter in front of the sensor to limit incoming light to the visible spectrum, to replicate what a human eye can see. But wherever I look on the internet, I always read about the filter in front of the sensor being an infrared filter. Wouldn't the filter also have to block out UV light? I couldn't find any useful information on this on the internet :/ Also, wouldn't an active filtering of UV light in front of the sensor render UV lens-filters useless?

UV is annoyingly present when doing landscapes and aerial photography. It records as a haze that blocks the clear view of distant mountains and it veils the land when imaged from high altitudes. A UV blocking filter can be very helpful under these circumstances. The UV filter and a cousin called a “Skylight” filter gained popularity. The “skylight” is tinted pink, so this UV filter also warmed up cool feeling blue-sky type vistas. Special note: The UV filter only benefits when the subject is distant and shrouded by water vapor. Camera store salesmen, eager to pad a sale, generally advised, a UV filter will protect your precious, costly lens. The popularity of the UV thus soared.

With the onset of the digital camera, the need to mount a UV filter diminished because electronic photography raises different issues. The imaging sensor requires trimming with filters or it will fail to deliver a faithful image. The surface of the digital sensor is covered with an array of tiny photosites. These capture the image, but the chances that artifacts with spoil it are high. Most noteworthy is image noise. This is akin to grain in film photography. There are a plethora of these annoying artifacts.

Enter the digital camera’s protective cover glass. The surface of the digital image sensor is fragile, it is covered by a flat glass overlay. This cover glass lends itself to have a dual purpose. Some subject types will image with bizarre results. These are called “demosaicing artifacts, often seen as a moiré. To avoid, the cover glass is also a optical low-pass filter better known as a anti-aliasing filter. This filter slightly blurs fine detail that is finer than the native resolution of the senor. Additionally the cover glass will act as an infrared filter that blocks these frequencies otherwise they will record as false colors

The UV continues to be sold and mounted to protect our precious lenses.

UV lens filters in the digital era have a different purpose than actively filtering UV light. They are to protect the front element from dust, fingerprints, etc. If the glass in front of the lens gets dirty, you are much safer cleaning a $50 UV filter than a $500 lens front element.

Digital sensors are typically insensitive to UV, so you don't need the UV filter to filter it out. Source: https://www.dpreview.com/articles/7333331953/should-you-use-a-uv-filter-on-your-lens which says:

However, digital sensors are generally rather insensitive to UV, so the problem doesn't arise to anything like the same extent.

In terms of filtration there is not only the IR filter, but also the RGB CFA (color filter array) that lies directly over the sensor. These are absorptive filters and only pass the wavelengths they are designed to. I.e. adequate UV filtration is provided by the RGB Bayer array (or similar).

I have read that silicon photodiodes (pixels) are more sensitive/reactive to IR than they are to UV. This is also probably due, at least in part, to the higher refractive index of shorter wavelengths.
Perhaps that is why the CFA alone is not sufficient for near IR but it is for UV.

In any case, the individual spectral response of a camera/sensor varies somewhat... some are more or less affacted by UV/IR even with their filtration.

Traditional metal-oxide "diodes" register UV in very low portions.
Here is a chart for "high UV sensetive" cmos: https://www.cameraiq.ru/data/image/QE%20MicroVista%20UV.JPG

Deep UV is ~270-330
Near UV is 330-400

As you can see, even special cmos can actually reach only small part of bellow-380 UV, while simpler ones actually reach only 400+, wich is visible light

So yes and no, cmos can see UV, but they are much more sensitive in visible part of spectrum.

In everyday life, while not being somewhere your oncologist wan't be happy to be, UV filter will not make any visible difference in your image, simple blend will make much more effect under bright sun. There are stories about shooting in the mountains, where people find some usage from this filters, but ok, even sample photos from filter-makers, are good both with and without filter :-)