This document discusses various types of discontinuities and limits that can occur in functions, including jump discontinuities where the left and right limits differ, point discontinuities where the limit at a single point does not exist, holes where the function is undefined within an interval, and vertical asymptotes where the function approaches positive or negative infinity at a single point. It also notes that a function can be said to not exist over an entire interval if its value is undefined for all values within that interval.