What is it called for the part of a hill where it discontinuously gets steeper?
As far as I am aware there is no single term to describe the sudden change in the gradient of a slope. Personally I would describe it simply as a sudden increase in slope. However there are many words to describe the landforms that occur when gradient sharply increase.

Terms such as cliff, escarpment and mountainside all describe the steep landform as whole. Whereas terms such as flank or face describe the side of the hill only with a steep gradient.

You might find this link particularly helpful:
https://www.macmillandictionary.com/thesaurus-category/british/parts-and-features-of-mountains-hills-and-cliffs