Which is correct: "anytime" or "any time"?

Is it "any time" or "anytime" or are those two things different?

One may write

Call me anytime

Call me any time

Call me at any time

but not

*Call me at anytime

because anytime is used as an adverb and not a noun.

Here is the explanation: "At" is a preposition, so it must be followed by a noun. The noun in the example is "time." "Any" is an adjective modifying the noun "time." It's like "Call me at four o'clock. Call me at night. Call me at any hour of the day or night." But "anytime" is an adverb, like "anyhow" or "anyway." It's like "Call me early" or "Call me occasionally." So why did the answer indicate that "Call me any time" is also acceptable? Because using "anytime" is more informal. The word does not even appear in the Webster's New World Dictionary of 1988. However, it does appear in the online dictionary (dictionary.reference.com). Evidently, the use of "anytime" has become more acceptable.