I can't pick a portion of math to call my favorite. All of math has had something "beautiful" to show, something "useful", in one way or another, to offer, in my experience.
I didn't always think so. When I was younger, I had a friend who asked me about this subject, in a van on the way back from a place called "Fermilab". I told her that I felt like math was just a necessary evil, almost punishingly complex. It was just the price of admission into any real conversation about physics - the, only set of "things" that I would respect as "real". I told her that what we called "math" was too messy. Too clunky. Too full of stuff to memorize, full of unmotivated tricks for geeks and barbaric approximations for engineers. I thought, surely there's a better way to get to and to understand answers about the world. She, on the other hand, called mathematics "the greatest monument to the mind of man." I said she was full of it.
Well, we were both wrong, I think. No, it's nothing so lame or vain as a monument. And if it is, at least surely not to us. But it is also as "real" as anything ever could or can be - maybe even the only "thing" there is, as the apparent source of all other things (depending on what you really mean, when you say, "math"). With this change of heart, I haven't since seen the study of math as being that of a shallow assortment of hard rules and puzzle-solving tricks, but as an invitation. THE invitation. In reply, I suggest you make all of it your favorite.