1. A function is continuous at a point if it satisfies three conditions: it is defined at that point, the limit of the function as it approaches the point exists, and the limit equals the value of the function at that point. 2. There are three types of discontinuities: removable discontinuity where the limit exists but does not equal the function value, jump discontinuity where the left and right limits do not match, and infinite discontinuity where the limit is infinity. 3. The document provides examples and explanations of continuity and different types of discontinuities in functions. It encourages the reader to check additional video resources for more information.