I don't think 24 hours a day is really considered a significant metric (at least not enough to warrant tracking it). The first seven day per week operation was the CBS affiliate in New York, in 1931, though CBS didn't operate the affiliate from 1934-1940.
I'm not able to find it easily, but I'm betting it was a broadcast station that was sent out via cable, not a cable-only station.
My best recommendation for getting a good answer is to call the Paley Center in NYC (formerly the Museum of Television and Radio.) They are very likely to be able to find it in their collection.
wor ch 9 in ny shortly after cbs However CBS NY was on most of the night from 1950's with the movies all night ie " THE LATE SHOW " THE LATE LATE SHOW & THE LATE LATE LATE SHOW " . CBS then went off the air around 5 am and started again at 6:15 WOR's LATE MOVIE I THINK WAS CALLED The Milkmans Matt.
Many local stations were 24/7 long before Ted Turner bought his first TV station, WTBS, in 1970. They all tended to be independent stations that ran movies. Most network affilated stations signed off for at least a few hours every night. First station I worked at was off the air from 1 a.m to 5 a.m.
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