How is 之 used in these sentences? 不能实现梦想的人都有一个共通之处，那就是：想要一样东西，却怎么也不愿意为之付出

I do not understand the usage of 之 here. What does 之处 and 为之付出 mean? Thank you.
The former 之 is equivalent to 的, but more often used than 的 when followed by a single character that would otherwise be expressed in a two-character word. In this case, it is followed by 处, which means “place” 地方 and by extension “aspect” 特征. Thus, the first part of the sentence means word-by-word "People who cannot realize their dreams are the same in one aspect(characteristic)," or more in more appropriate English "Those who fail to realize their dreams have one thing in common,"

The latter 之 has a different meaning, as is common in classical Chinese (文言). It is the third person pronoun in the objective case, meaning that it should only be used in place of 他/她/它 when the thing in question is the object of the sentence, much like "him" instead of "he" in English. Therefore, 为之付出 means to "pay effort for it".

More examples of 之 as a third person objective pronoun


授之以渔 (classical) "Teach them with fishing" = "Teach them the art of fishing."
将之磨碎 “grind it into chunks/powder/etc", equivalent to 把它磨碎， note that 将 is always used instead of 把 before 之.
有过则改之 (proverb) "Have a fault, then change it" = "If you made a fault, mend it".
投我以桃，报之以李 (proverb) "Give me with peaches, return them with plums" = "They give me peaches, I return them the favor with plums."