How long have commercial aircraft been continuously in the sky?
This is a great question and I will try to answer as I think it was intended. Based on reasonable assumptions, I can estimate that at least one commercial passenger flight has been in the air somewhere in the world continuously since the early 1930s.

A key help in answering the question is that passenger aviation grew very rapidly throughout the mid-20th century. Moreover, the chance probability of having zero flights in the air at some moment is mathematically very sensitive to the overall rate of flights. Thus, the historical window in which an "empty sky" is plausible would close quickly. Past some point, the overwhelming growth makes it virtually impossible that the skies would ever be devoid of commercial flights again, unless some event stopped them in sync almost everywhere in the world.

What about World War II?

In North America, despite the diversion of a significant fraction of people and resources to the war effort, many commercial passenger flights continued. Schedules in the early 1940s showed numerous routes operating at all hours of the day and night. So much growth had occurred in the 1930s that even the dent from the war left the industry well ahead. (This may well have also occurred in Australia, another developed region mostly spared the direct violence of the war domestically.)

What about 9/11?

Although US and Canadian airspace was closed after the attacks, flights continued within the rest of the world -- again making a dent but not undoing decades of growth.

So why the early 1930s?

Due to the aforementioned rapid growth, we will be pretty close if we merely get the orders of magnitude right. Assume a typical flight length of 2 hours or 200 miles in this era. Also assume that flights were uniformly distributed over a time period (say a year) -- equally likely to be scheduled at any time 24/7/365. Even within one region, flights operated day and night, and superimposing the various global time zones (even just North America and Europe) further homogenizes the flight schedules.

If we can estimate the average number M of commercial flights in the air at any moment, we can model the fluctuations statistically as a Poisson distribution. According to the Poisson distribution, the probability of finding zero flights at a given moment is e^(-M), where e ~ 2.7. This result is fairly robust; for example, if we take a different model where there are 2*M planes in the global airline fleet, each of which is flying half the time (thus M are in the air on average), then the chance that no plane is flying at a given moment is (1/2)^(2*M) = 4^(-M). The key result is that the probability decreases exponentially with M.

Roughly every 2 hours, the flights "turn over" (current flights have landed and a new set has taken off), giving us a new sample from the probability distribution. During a year, we get about 4,000 samples, and the probability that we see zero flights at least once is roughly 4,000 * e^(-M). Of course, M is increasing from one year to the next as the industry grows. We conclude that once M surpasses about 15, the probability of ever again having the sky globally devoid of commercial flights is negligible.

In the year 1930, there was already a noticeable passenger airline industry in the US (at least 6,000 passengers flown, which is likely very low because another source says over 300,000) and in Europe (10,000 in Italy, 18,000 in the Netherlands, not to mention the UK, France, and Germany).

If we estimate a worldwide total of >100,000 passengers for the year, and consider that a typical ticket was for several flight segments, say 1,000 miles, then we get >100 million passenger-miles in 1930. As further confirmation, the US alone is quoted at 84 million passenger-miles that year.

The planes were very small and load factors were low, so the typical number of passengers on board was likely around 3. Thus, passenger flights worldwide traveled at least tens of millions of miles in 1930 (compare 37 million revenue miles for US airlines, which may however include some cargo flights without passengers).

At an effective speed of 100 miles per hour, the total flight time was at least hundreds of thousands of hours. With the year containing about 9,000 hours, the sought average number M of planes in the air at once would be dozens. By the statistical reasoning given previously, this is the point at which the "empty sky" probability becomes negligible. And if we are off by say a factor of 2 in estimating M (e.g., due to seasonal variations), this corresponds to just a few years of growth.