By ANTHONY DePALMA

Published: April 12, 1987

INVINCIBLE COURT is a vintage apartment building of 1920's Harlem, a brick dowager with a dainty sitting area in the lobby and 53 brass mail boxes that still close and lock.

A one-bedroom apartment on the third floor recently was vacant and available for rent. The parquet floors had long since gone gray and all the wood moldings were muffled with paint. A piece of the kitchen ceiling had fallen and needed to be repaired. The living-room windows opened onto a broom-clean concrete courtyard.

The asking rent was $243.51 a month, heat included, but what makes that apartment unusual is not the low rent. Despite the common perception that all rents in the city are sky high, the officially recognized median rent of all occupied apartments in the five boroughs was only $330 in 1984, the last year the city counted. What was unusual about apartment 3-E was that it was available at that low rent, making it an increasingly rare and, some say, nearly impossible-to-find commodity today.

When such apartments are vacated in any but the most-distressed neighborhoods, rents rise, sometimes sharply, and legally, under the state's regulation system. The economics of operating low-end residential buildings in the city give landlords what they say are compelling reasons for raising rents as far and as fast as they can.

The risks of investing in such buildings, including the inability or unwillingness of some tenants to pay rent, are such that few landlords are willing to operate them for the small returns they can earn on rental income. Rather, they seek quick capital gains on the rising value of the land by selling the property at a multiple of a maximized rent roll. In depressed neighborhoods that multiple may be as low as three. It gentrifying neighborhoods, it can be driven as high as 10.

What this means in the ultimate sense is that almost no one in New York City is in the business of providing housing at low cost. In an economic sense, it may even be impossible for a landlord to provide decent housing at the median rent of $330.

When the city runs a foreclosed building, it pays no taxes, carries no mortgage interest and seeks no profit, but even so, the Department of Housing Preservation and Development finds, on average, that it costs $270 to $337 a month to operate an apartment. When taxes, interest and a modest return are added in, the cost of operating such an apartment, owners estimate, is $465 a month. Yet only about half of the city's households can afford rents that high if rent is to represent no more than 30 percent of their income.

''I've been looking for nine months now, all over New York,'' said Kimala Days, a teacher's assistant at the United Federation of Black Community Organizations' day care center in Harlem. Ms. Days recently called to look at an apartment in the Invincible Court. It had only one bedroom, and she said she had hoped to find a two-bedroom apartment for herself, her husband and their son that cost about $400 a month. Until she comes up with something, she will continue to live with her mother in the Bronx.

Finding that apartment is a daunting task. Ms. Day cannot rely on newspaper ads or referral services. In the rare instances where an apartment below $330 is advertised, it almost invariably is a come-on. Recently, a Bronx agency ran ads for apartments renting for $300 to $695. When called about them, the owner admitted the lower-priced units did not exist. She said she had intended to change the bottom rent to $375, though she had nothing available at that price either. AND yet, bottom-of-the-market apartments do exist, in sizable numbers according to official estimates. The city is getting ready to conduct a new housing and vacancy survey, and the median rent -which has risen by over 50 percent since 1978 - is expected to show another sharp increase. But for now, the most recent figures on rent come from the 1984 report. Then, the median contract rent - what current tenants actually pay - was $330, including utilities. That means half of the 1.9 million apartments in New York - 950,000 residences - rented for less than that and half for more.

''The housing market in New York is very fragmented,'' said Peter Marcuse, a professor of Urban Planning at Columbia University. ''What happens south of 96th Street is very different from what happens in central Harlem or in the South Bronx. But since many of us just move south of 96th Street, we really see only a part of the picture.''

Manhattan south of 96th Street generally conforms to the common perception of New York as a high rent district. There the median rent is the highest in the city at $476, with asking rents often above $1,000.

Indeed, Edward Carter, a research assistant in the film department at the Museum of Modern Art, said that while he has been unable to find a studio apartment he could afford by himself on his $14,600 a year salary, one of his coworkers pays less than $100 a month for a rent-controlled Manhattan apartment he has occupied since 1968.