Would a discount broker by any other name still sell stocks and bonds so cheap?

Keith Aufhauser thinks so. Mr. Aufhauser is president of K. Aufhauser & Co., a New York brokerage firm that has a reputation for low prices.

But don't call it a ``deep-discount'' broker. ``We are a full-service firm which performs transactions at a low cost,'' Mr. Aufhauser says.

Generally speaking, deep discounters are simply the firms that charge the lowest commissions. But there's really no formal definition. ``If they say they are, they are,'' says John Markese, president of the American Association of Individual Investors.

It used to be a lot easier. If the firm did your trades for a cut-rate commission with no frills included, you were dealing with a ``discount broker.'' But in recent years, the waters have muddied.

The ``Big Three'' discounters _ Fidelity Investments' Fidelity Brokerage Services, Quick & Reilly Group's Quick & Reilly Inc. and Charles Schwab & Co. _ have rolled out a rainbow of services that make them look ever more like full-service firms.

Meanwhile, a host of new firms has emerged, boasting prices that are typically lower than those charged by the Big Three while offering services that range from bare-bones to nearly deluxe.

Some of the most recent entrants into the increasingly crowded field advertise rates as low as 1 1/2 cents a share, marking the birth of a new subspecies. Mark Coler, president of Mercer Inc., a New York company that keeps tabs on discount brokers (and deep discounters, too), calls these low-ball firms ``deep-deep discounters.''

Like Mr. Aufhauser, though, not everybody who might be considered a deep discounter relishes the label.

``It's sort of tacky,'' says Everett Lang, president of Sherwood Group Inc.'s National Discount Brokers, a New York firm. Still, Mr. Lang says he takes no offense.

Then there's the view at Wall Street Equities Inc. of New York, a firm that's happy to wear the deep-discount tag. ``It's fine with me. I'm providing a valuable service to the public,'' says Edward DeFelice, Jr., vice president of the firm.

Mr. DeFelice, a native of the Paramus, N.J., area, adds: ``Where I'm from, there's an old saying. `Call me anything you want. Just don't call me late for dinner.'''