Creating a color gradient on a specific text object part

That's right, Ton.

It is based on pattern fills and may (more often than not) require some modifications.

The advantage is that you can apply different gradient fills to single words or characters. Not only one gradient as per your example.

But frankly speaking, I think that most of the time it is rather appropriate to just convert the live type to paths and then apply the desired gradients.

I'd rather assume that no one is going to type an extensive 700 pages book using these live type gradient tricks.

Here you can download an Illustrator file with another approach.

https://drive.google.com/file/d/1HnYo9PR_KAxSDt8RPCB4WZQYLgctKSio/view?usp=sharing

Rita,

If you wish to have a more irregular kind of colour transition(s), you can also place the live Type on top of a larger path with the exact desired colour transition (for the screenshot solid white at the top and a mottled gradient at the bottom, as on the M), then select them both and apply Object > Clipping Mask > Make. You can adjust the exact positioning after the Clipping Mask/clipping set has been established by selecting either and moving it about  (you can select either easily through the expanded Layer).

This is pure genius, Ton. All these years… what a thrill to discover a secret like this. Thank you, as always.

A text object has a single set of appearance attributes, if that's what you're asking. However, appearances are very versatile things. It may or may not be possible to devise a method depending on the specifics of what you want -- if you want to go into specifics.

Without seeing what kind of gradient on what specific part, I would say yes.