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

Verified step by step guidance

1. <p>1. **Understand the Function**: The function \( p(t) \) represents the number of points scored by a basketball player after \( t \) minutes of a game. This is a real-world scenario where the function is defined in terms of time and points scored.</p>
2. <p>2. **Identify the Nature of the Function**: In this context, \( p(t) \) is a step function. A step function is a piecewise constant function that jumps from one value to another without taking on intermediate values. In a basketball game, points are scored in discrete amounts (e.g., 2 points for a field goal, 3 points for a three-pointer), and the score changes instantaneously at the moment a basket is made.</p>
3. <p>3. **Determine Continuity**: A function is continuous if there are no breaks, jumps, or holes in its graph. For \( p(t) \), since it is a step function, it will have jumps at the moments when points are scored. Therefore, \( p(t) \) is not continuous because it has discontinuities at the times when the player scores.</p>
4. <p>4. **Consider the Domain**: The domain of \( p(t) \) is the duration of the basketball game, typically from \( t = 0 \) to the end of the game. Within this domain, the function is defined, but it is not continuous due to the nature of scoring in discrete steps.</p>
5. <p>5. **Conclusion**: Conclude that \( p(t) \) is not continuous for all values in its domain because it is a step function with jumps at the points where the player scores.</p>

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.