The phrase "keep a constant eye" is correct and usable in written English.
You can use it to refer to the act of monitoring something closely over a period of time. For example, "The teacher kept a constant eye on the children as they played in the playground."
In return for their help, I try to repay them by keeping them apprised of news developments (I keep a constant eye on wire service reports) and, occasionally, going to news conferences.
The prudent owner must keep a constant eye on wastefulness.
The notion that you get ahead faster by switching companies is reinforced by career counselors, who advise people to keep a constant eye on outside opportunities.
We avoid overcrowding the pan, consider when to add each component, move things from pan to oven, when necessary, and keep a constant eye on the temperature.
Long ago, to prepare for the anticipated crowds, more than a dozen monitors were hired to keep a constant eye on the well-being of flora and fauna.
"The mayor always said that we have to keep a constant eye on whether we can afford these tax cuts, and based on where we are right now, they're still alive and moving forward," the aide said.
Calliope keeps a constant eye on the other make-out session.
He was not averse to risk; he kept a constant eye out for new projects and acquisitions.
When Nik Deogun was running the financial coverage of The Wall Street Journal during the crisis in 2008, he kept a constant eye on CNBC as the events played out on television.
They don't want to be their own doctors, train their own doctors or keep a constant wary eye on their doctors.
Give away boxy suits and file standoffish square nails into rounded ones, says image consultant Sandy Dumont, adding that it always helps to flash a bright smile and keep a constant twinkle in your eye.
