Which of the following functions are continuous for all values in their domain? Justify your answers.


d. p(t)=number of points scored by a basketball player after t minutes of a basketball game

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over its entire domain, it must be continuous at every point in that domain. This means there are no breaks, jumps, or asymptotes in the function's graph.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Understanding the domain is crucial for determining continuity, as a function may be continuous within its domain but not defined outside of it. For example, a function that involves division cannot include values that make the denominator zero.

Real-World Applications of Functions

In real-world scenarios, such as the scoring of a basketball player over time, functions can model behaviors that are inherently continuous. For instance, the number of points scored can be represented as a function of time, where the function is continuous as long as points are scored in a consistent manner without sudden jumps, reflecting the natural flow of the game.