By Anora Mahmudova and Carla Mozee, MarketWatch

ISM services index falls in December

NEW YORK (MarketWatch) -- U.S. stocks turned down amid choppy early trading that had suggested a more upbeat start to Tuesday trade, as oil prices continue to slump.

Investors are maintaining a cautious posture after a brutal sell-off on Monday, which saw the closely watched Dow shed 331 points.

On Tuesday, investors were scouring for safe industries such as utilities, health care and consumer staples, as well as U.S. Treasurys, pushing yields to seven-month lows.

The S&P 500 (SPX) was roughly flat with defensive sectors, such as utilities, health care and consumer staples moving higher earlier in the day. Energy stocks that fell 4% on Monday were also relatively higher.

The Dow Jones Industrial Average (DJI) inched lower. The Nasdaq Composite (RIXF) dipped in and out of positive territory and was heading lower.

10-year Treasurys rallied, pushing the yield 5 basis points to below 2% for the first time in seven months., indicating that investors are seeking havens like U.S. government debt. Meanwhile, oil prices dropped another 2%.

Kate Warne, investment strategist at Edward Jones, said that Monday's selloff was an overreaction to uncertainty surrounding the eurozone economy as Greece looks likely to exit the union, while a slide in oil prices stoked further worries about stagnating global economic growth outside of the U.S.

"Stable stock prices today is good news and would mean that investors are assessing if the U.S. economy still remains an island of growth. In the long-term, lower oil prices is a positive to major economies, but until oil finds a bottom, there will be short-term fear over global growth, Warne said.

In economic news, companies in the U.S. service sector such as retail and real estate grew at a slower but still-healthy pace in December, according to survey of senior executives. Orders for goods produced in U.S. factories fell 0.7% in November, hit by transportation equipment, matching the forecast from MarketWatch-polled economists, according to government data released Tuesday.

Analysts also pointed that investors would be weary of making big bets ahead of the important jobs report on Friday, which will shed light on how well the U.S. economy is weathering amid global weakness.

On Monday, U.S. stock markets took a battering as crude-oil prices slid again and as the dollar (DXY) surged. The S&P 500 and Dow marked their biggest one-day losses in three months.

"One would question whether this is 'Turnaround' Tuesday or simply a pause before the next leg down," said Brenda Kelly, chief market strategist at IG, in emailed comments. "The plunge in equities yesterday was unexpectedly extreme, so we may be witnessing some profit-taking and an element of bargain-hunting."

Overall, "we are still in a low-interest-rate environment, and as has been the case for the past few months, risky assets provide a yield that cannot be gleaned elsewhere," she wrote.

Crude-oil prices extended losses on Tuesday, with U.S. oil futures (CLG5) down 1.8%, trading just below $49 a barrel. Brent futures were down 1.2%, off intraday lows.

"The economic conditions that oil faces continue to be aggressively against the commodity, making buyers extremely hesitant to even consider entering long positions," said Jameel Ahmad, chief market analyst at FXTM, in a Tuesday note. "There are also suspicions that as oil companies struggle to adapt to lower profits, there could be mergers and acquisitions taking place in the future, and this will weigh on investor sentiment. What's the answer to this equation? Bearish moves for oil."

Stocks to Watch: Verizon Communications Inc. (VZ) has approached AOL Inc.(AOL) about a possible acquisition or joint venture, Bloomberg reported Monday, citing people with knowledge of the matter. Shares in AOL jumped.

Cyberonics Inc.(CYBX) shares fell after the medical-device maker said an appeals board supported a 2007 decision by the Centers for Medicare and Medicaid that denied coverage of the company's pacemaker-like device for treating depression.

Coach Inc. (COH) shares were up slightly after the company said it's buying luxury footwear brand Stuart Weitzman from private equity firm Sycamore Partners, in a deal valued at up to $574 million.

Nielsen N.V. (NLSN) shares fell after financial-news network CNBC said it will no longer rely on the TV-ratings specialist to measure its daytime audience, beginning later this year.

(Read more: Movers & Shakers column http://www.marketwatch.com/story/cyberonics-micron-shares-in-focus-tuesday-2015-01-06.).

Other markets: Meanwhile, the yield on the 10-year Treasury fell below 2%, to 1.97%, according to Tradeweb. Yields have been hitting lows not seen in almost two years, discounting the mini flash-crash in mid-October, indicating investors are looking to bonds as a safe haven.

In Asia, the Nikkei fell 3% to post its biggest one-day loss in nearly 10 months, and Indian shares plummeted 3.1%, the steepest loss since September 2013. European stocks turned higher, pushing back from a loss that followed a clutch of lackluster economic data.

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