The stock market continued its summer downturn today, although fears of a Monday disaster were muted as Cities Service began implementing a program designed to stabilize the price of its stock, tattered after Gulf Oil backed out of a takeover effort last week.

In heavy trading, Cities Service stock fell 6 7/8 on the New York Stock Exchange to 30 3/8, while the Dow Jones Industrial Average fell by almost 4 points to 780.35. The industrial average has slipped by more than 53 points since July 20 and analysts unanimously predict the slide will continue through the summer.

Cities Service reacted to Gulf's action by authorizing the purchase of up to 20 million shares of its own stock and has begun the search for a new merger partner. The company said if those moves fail it might consider liquidation, which would almost certainly produce a higher price for the stock than yesterday's figure.

"At 4 p.m. Friday, the street was thinking disaster," commented Larry Wachtel, first vice president for research at Bache Halsey Stuart Shields Inc. "Today there was a collective sign of relief when we showed up in the morning. Cities Service took most of the panic out of the market."

The selloff had been set Friday because many brokerage and other trading firms had staked out large positions in Cities Service stock. With the market edgy over the Penn Square National bank collapse, the Drysdale Government Securities fiasco, and mounting financial pressure on corporations, investment strategists seemingly tremble with each new shock wave.

Losses were broad and continuing today. Particularly hard hit was the American Stock Exchange, which is heavily dependent on energy stocks. About 45 percent of the capitalization of Amex firms is in oil and other energy-related stocks.

The Amex index fell to 240.36, down 4.63 points, while volume rose to 4.3 million shares from 3.2 million Friday and the average price per share fell 18 cents. Only 105 Amex stocks advanced today, while 500 declined.

The picture was only slightly better on the New York Stock Exchange, where 402 stocks rose and 1,107 declined. Volume rose to 54.9 million shares up from 48.6 million. Over the counter stocks also slipped, as the NASDAQ composite index fell 2.21 to 161.33. Only 130 over the counter stocks rose, while 1,171 declined.