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The phrase "are continually on" is correct and usable in written English.
It can be used to describe a state of being consistently active or engaged in something over a period of time.
Example: "The team members are continually on the lookout for new opportunities to improve their skills."
Alternatives: "are constantly engaged in" or "are always focused on".
Dictionary
are continually on
adverb
In a continual manner; non-stop.
Exact(14)
As one of them put it: "If you look like you are continually on the point of shouting, and a bit of a nutter, you can be very effective at getting things to happen.
To be sure, inferences must be made from the categories of one's metaphysics, but it is the categories themselves that are continually on trial.
Because aggregations are continually on the move, the leks are only temporary.
As demand for Internet service grows, communications carriers are continually on the lookout for faster routers.
The Kiwi challengers seem to have made most progress in keeping their boat stable, but they are continually on a knife-edge.
Under the new approach, rather than having artificially marked-up items that are continually on sale, it will try three price levels: a regular everyday price, a special monthly price for seasonal items and a clearance price.
Similar(41)
But that role was always an 18-month role and the work will be continually on-going".
"The basic one, when you're continually on the move," he said, "is getting fed.
"The businessman of the present day must be continually on the jump he must use the telegraph," grumbled one New York merchant in 1868.
Labelled a whore, she is continually on the move towards normality and solace.
The coalition government they created is continually on the verge of collapse — not exactly an encouraging sign for the people of Ivory Coast.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com