N4
それでも
But still, And yet, Even so, Nevertheless
今日は雨が降るよ、それでも釣りに行くの?
It's going to rain today, even with that, are you still going fishing?
Explanation
「それでも」 means 'even so' or 'and yet', at the start of the second sentence. The first sentence mentions an unfavourable situation, an obstacle or someone's advice, and the second says the subject goes ahead anyway: 「大雨が降っている。それでも、彼は出かけた」 (It's pouring. Even so, he went out). It shows strong contrast and stubbornness. In conversation it can be a challenge: 「それでも〜の?」 (Even so, will you …?).
Mixed up with
Compare the connectors: 「それでも」 means 'even so, still' (insisting on it); 「しかし/けれども」 is an ordinary 'but' (A, but B); 「それにしても」 means 'even allowing for that, it's too …'; 「だからといって」 often goes with a negative: 'that doesn't mean …'.