CHICAGO -- Eugene Sawyer, a soft-spoken former chemistry teacher, has faced an almost impossible task during his first week as acting mayor of Chicago.

He has been forced to try to demonstrate that he has a political base and a backbone. The odds seemed against him when he was elected to the job a week ago by the City Council.

Sawyer is a black elected by white aldermen in less than auspicous circumstances. To gain credibility among blacks, he has to find a way to show he did not sell out their interests. To do this, Sawyer risks alienating the whites who elected him, and being left without any support.

It probably will be months before Sawyer, Chicago's second black mayor, can show skeptics whether he is tough enough to govern the nation's third-largest city.

But he has made some progress toward establishing a shaky political base in this racially divided city during his first days in office. Sawyer has had a great deal of help from an unlikely ally, presidential candidate Jesse L. Jackson.

Jackson tacitly supported Sawyer's chief rival, Alderman Timothy C. Evans, to succeed the late Harold Washington, Chicago's first black mayor.

Jackson, embarrassed by his failure to succeed as a "kingmaker" or "peacemaker" in his hometown following Washington's death, staged a series of well-publicized photo opportunities with Sawyer this week.

Jackson has stopped short of endorsing Sawyer, a longtime South Side ward boss, during these sessions, but he has urged a "cooling of tempers" and suggested that Evans stop efforts to taint Sawyer's election.

"We must judge a new mayor by his appointments, jobs, legislation, contracts. And time becomes the great healer of all things," Jackson told one group of community leaders.

Sawyer has returned the favor by endorsing Jackson's bid for the 1988 Democratic presidential nomination and calling the civil rights leader "the next president of the United States."

"Rev. Jackson is a tremendous galvanizing influence," said Alderman Danny Davis. "He has been very helpful to the new mayor in the black community."

Many of the blacks, Hispanics and liberal whites who formed the coalition that elected Washington to two terms remain highly skeptical, however. "I'm waiting to see if he "Few people will ever face the kind of pressure he'll face in the next few months."

-- Paul Green, Chicago academic

will move away from the individuals who put him in office," said Alderman Luis Gutierrez, an Evans supporter.

Grass-roots groups have organized opposition to at least two black aldermen who voted for Sawyer, and the new mayor may face opposition in his own ward when he runs for reelection as a Democratic Party committeeman next March.

"He has a tough and uphill battle ahead of him," said Paul Green, an academic specializing in Chicago politics. "He has enormous problems ahead of him. Few people will ever face the kind of pressure he'll face in the next few months."

These problems are compounded by the fact that Sawyer has no safety net to catch him should he stumble.

Before Harold Washington died two weeks ago, Sawyer was virtually unknown to most voters outside his South Side 6th Ward.

All that most people know about him now is what they have seen on their television screens in recent weeks. The first impressions were not a pretty sight.

Sawyer, who grew up poor and black in rural Alabama, was elected acting mayor at 4 a.m. a week ago by a group of white aldermen in one of the most raucous Chicago City Council meetings in history. His opponents berated him for hours on TV for allying with "wolves," "vultures" and "thieves." Thousands of demonstrators chanted "Uncle Tom Sawyer" and some of his supporters complained of receiving death threats.

Sawyer was unnerved. He wavered for hours over whether he would take the job. At one point, his knees buckled and he had to be helped out of the council chambers.

"He had to be held up and shoved into the seat in the wee hours of the morning presumably after he became too weak to resist. No telling when his will might blow in the wind again," The Chicago Tribune editorialized. "Having to jump-start the mayor several times before each tough decision does not bode well for the cold winter to come."

Sawyer's first big test is a $84 milion property tax increase that Washington had proposed to balance the city budget. Sawyer endorsed the increase the day after he became mayor, but one of his chief white supporters, Alderman Edward M. Burke, promptly vowed he would led a battle against the increase.

Joe Novak, a political consultant, expects to see more such moves from Sawyer. "He has to drift over to Washington's reform agenda if he wants to survive," Novak said. "The people who put him into office are going to be disappointed that they can't go back to their communities and say we got this or that. This will result in more choas and fighting. The only real winner in all this will be the Republican Party."