Most commentators consider the adverb sure to be something less than completely standard; surely is usually recommended as a substitute. Our current evidence shows, however, that sure and surely have become differentiated in use. Sure is used in much more informal contexts than surely. It is used as a simple intensive <I can never know how much I bored her, but, be certain, she sure amused me — Norman Mailer> and, because it connotes strong affirmation, it is used when the speaker or writer expects to be agreed with <it's a moot point whether politicians are less venal than in Twain's day. But they're sure as the devil more intrusive — Alan Abelson><he sure gets them to play — D. S. Looney>. Surely, like sure, is used as a simple intensive <I surely don't want to leave the impression that I had an unhappy childhood — E. C. Welsh> but it occurs in more formal contexts than sure. Unlike sure it may be used neutrally—the reader or hearer may or may not agree <it would surely be possible, within a few years, to program a computer to construct a grammar — Noam Chomsky> and it is often used when the writer is trying to persuade <surely a book on the avant-garde cannot be so conventional — Karl Shapiro>.
Examples of SURELY
She answered quickly and surely.
He will surely be missed.
This is surely the best dessert you have ever made.
Surely you must admit that it was a good decision.