The notation "C[0,2]" refers to the set of all continuous functions defined on the closed interval from 0 to 2. This means that any function included in this set must be continuous for every point between 0 and 2, including the endpoints. Continuous functions do not have any breaks, jumps, or holes in their graphs within this interval. In mathematical analysis, continuous functions are important because they exhibit predictable behavior. For example, if you take any two points within the interval 0, 2, the function will not suddenly change values, making it easier to analyze and apply various mathematical techniques, such as integration and differentiation.