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
之 has many use cases, these two are used to denote a property and an article like "it", or "that".
The literal translation for these 2 4-character phrase.
共通之处 -- something in common / somewhere in common
为之付出 devote for it
In the case of "共通之处", "之" seems to contribute no meaning and can be removed, since "共通处" could be used to mean also the same thing. This can happen sometimes when the speaker/writer is just adding the character as a structure support, and/or maybe to disambiguate. Since 4-character idioms are very commonly used like units of sentence making.