Short answers, pulled from the story.
The continuity thesis posits that modern mechanics and physics proceeded from doctrines professed in medieval schools through an uninterrupted series of improvements. Pierre Duhem argued that the Roman Catholic Church fostered Western scientific development rather than hindering it.
George Sarton became a notable supporter of continuity arguments during his active years between 1884 and 1956. He stressed historical continuity across time periods and claimed science had to be introduced to Western culture twice.
Nicole Oresme invented graphs and performed calculations involving probability during the twelfth century. This period represents the real renaissance according to James Franklin who challenged the idea of a European Renaissance as a myth.
George Saliba expanded on these critiques in 1999 when reviewing Toby E. Huff's work on Islam, China and the West. He questioned whether Ptolemy or Copernicus were closer to modern science than Chinese or Maya astronomers.
Edward Grant identified four factors combining to create origins of modern science within medieval times. The plague of 1348, 1350 killed a third of people in Europe and contributed to decline in some branches of science between Oresme and Copernicus.