But the question itself strikes me as a bit odd, in an age when mislabeled or unlabled internet content should probably be regarded as suspect/undesireable/dangerous.

OP's test for existance tells the name (and -- in some cases -- the nominal file.typ) of the target of the link. The answers above tell how to find out the actual type of file whether or not (OP's case) an extension is provided on the server.

OTOH, were one to rely on a browser, clicking a mislabeled link might provide perhaps as little info as "binary" (try this on a MSWord doc mislabeled as doc.foo, with FF under linux); perhaps misleading info on the actual type (content) of the file (try opening a .pdf mislabeled as an .xls, under w32).

Is there an X/Y problem here or am I missing some reasonable basis for the question?