At the beginning of the century.
In the beginning of the century.
How to clearly distinguish when to use at, or in?
In general, "at" marks a spot and "in" marks a space. Obviously, there are idiomatic exceptions, but this really isn't one. The beginning of a period of time is a "spot", the period of time itself is rather "spacious".
To answer the question: I would never say "in the beginning of the century". I think Peter Eisntraut's argument is essential correct: "the beginning of the century" is notionally a point, not a period (even though in practice, "at the beginning of the century" may in context cover a period of several years).
In fact the only instance I can think of of "In the beginning" is the opening of St John's Gospel. That phrase is now archaic, and would not be used except in imitation of that specific use.
"In ~" is Just Plain Wrong. Acceptable usages would be "at the beginning of the century" or "in the early years [etc.] of the century".
"In the beginning" as a collocation does not take a referent - it refers to some (contextually defined) absolute start (the beginning of the story, the creation of the world, etc.). It is familiar as the opening words of Genesis and of the gospel of John in the King James Bible.
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