When I was eighteen and learned how to drive a car, all cars already had fully synchronized transmissions and you could shift gears directly. But it felt sporty todo double clutching. When I did it the first time in driving school the driving teacher almost exploded. He thought there would be more valuable things to think about than reducing wear of the clutch. Actually I have never owned a car with an unsynchronized transmission, but I owned quite a number of cars suffering from weak synchromesh. and some earlier cars had unsynchronized first gears.
Recently I had lunch with a guy doing business around oldtimer and classic cars. And he surprised me with saying that he doesn't know what double clutching is. Yes, he was quite a bit younger than me, but still! By the way, what is called double clutching in English, we call "Zwischengas" in German. Translating this word by word leads to "operating the throttle in between". Funny enough the shifting up with double clutching is called "Doppelkuppeln", which means exactly the same what the English term says. And this also gave the name to the new transmissions, such as the PDK of Porsche or the DSG of Volkswagen. There have been debates on whether it's still useful to apply double clutching in modern cars. The answer is yes, and some modern robotized transmissions and sequential gearboxes even do it by themselves - thanks to electronics.
If you want to know more about double clutching, there are some good articles here and here. And thanks to YouTube you can also see it in full motion and color for example here.
By the way, the picture shows my Grantura's clutch. It would have taken more than a bit of double clutching to save this one.