I hear often in anime and is translated as "but". And normally, it's at the beginning of the sentence. But I wonder if it used outside of the TV/movie world. For instance, きみ, I hear very often in anime, but I've been told it's not used in everyday speaking.
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Ah, I just noticed we don't use だが very often in daily conversation. I think we often use だが in these three patterns (normally in written style):
I personally don't use it when I talk, maybe because it sounds rather masculine I guess. I think older, say middle-aged men (like my dad, my uncle, and my boss) say that sometimes. I think I've heard my dad say "だがしかし、あれだな～..." while watching TV or reading a newspaper and my boss say to my colleagues "だがねえ、きみ..."
As for きみ, I don't use that either, but I think my boss sometimes calls my colleagues きみ, and I've heard many of my teachers/professors call their students きみ at school/college.
が is common at the end of a sentence or to connect two sentences into one, and it may mean something like "but" or "although". It's difficult to say without more context, but I'm guessing the だ you speak of is the short form です, in which case I see no problem. (although I more often hear だけど (and ですが) than だが)
Lot of people have a different opinion on Japanese language. Like I personally think that きみ is used in everyday conversation, you just need the right environment.
だが is not used much but it is still used and it is not weird to use it if it matches the way you speak. It depends on the person's speaking level/style a lot.
It is written that だが is literary language and then, not really used by woman.