We know of two technologies which underwent a discontinuous jump in improvement equivalent to more than several decades at previous rates. These are explosives and superconductors. We know of two more cases of smaller discontinuities, and several promising cases to investigate further. These findings follow a short preliminary investigation, and are unlikely to be near exhaustive.
List of cases
Main article: Discontinuity from nuclear weapons
The development of nuclear weapons is a famous example of discontinuous technological progress. Nuclear weapons created a sharp discontinuity in the maximum ‘relative effectiveness‘ of explosives (that is, energy released per mass). The relative effectiveness of early nuclear weapons were 2,000 times greater than their predecessors, while previously relative effectiveness had grown by a factor of less than four over the past thousand years. Thus the jump represented around eleven doublings, or more than 6000 years of progress at prior rates. In terms of cost-effectiveness however, nuclear weapons do not appear to represent a substantial discontinuity.
High temperature superconductors
Superconductors were discovered in 1911. Until 1986 the maximum temperature for superconducting behavior had gradually risen from around 4K to less than 30K (see figure below). Theories at the time predicted that 30K was an upper limit. In 1986 a new class of ceramics were discovered to allow superconducting behavior at higher temperatures: above 80K, and within seven years, above 130K (see figure below). Thus the maximum temperature for superconducting jumped by around twice the distance it had moved in its entire 75 year history.
Jet propelled vehicles
Jet propelled vehicles arrived in the early 1960s, and according to Wikipedia’s list of land speed records, appear to have increased the land speed record from 407mph to 601mph between 1963 and 1965. The sharpest increase in their records was between November 2 and November 16 1965, during which the record went from 555mph to 601mph. This makes the absolute progress between 1963 and 1965 roughly equal to that in the 34 years beforehand (according to the Wikipedia records). At the rates of progress just prior to this change, it would have taken very much longer: the land speed record for wheel-driven cars had slowed down substantially (see figure below, or the Wikipedia page).
Fairey Delta 2
According to this data, the 1956 record, set in a Fairey Delta-2, increased the maximum speed by a factor of 1.4. This is around as much progress (in terms of multiplicative factors) as was made in the previous 11-17 years (since 1939-45). This is also roughly how long it took to make the previous absolute increase of 300mph.
The 1965 record, set in a Lockheed YF-12A, represented maximum speed being multiplied by 1.2. The previous factor of 1.2 took 7-8 years (since 1957-8) to produce, as did the previous 404mph. Thus this is a fairly small discontinuity