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


b. n(t)=number of quarters needed to park legally in a metered parking space for t minutes

Verified step by step guidance

1. Step 1: Understand the function n(t). The function n(t) represents the number of quarters needed to park legally in a metered parking space for t minutes. This is a real-world scenario where the function outputs discrete values (whole numbers) based on the input time t.
2. Step 2: Consider the nature of the function. Since n(t) outputs the number of quarters, it is a step function. A step function is a piecewise constant function that jumps from one value to another without taking on any intermediate values.
3. Step 3: Analyze the continuity of step functions. A function is continuous at a point if there is no interruption or jump at that point. For step functions, there are jumps at certain points (e.g., when the number of quarters needed changes), which means they are not continuous at those points.
4. Step 4: Determine the continuity of n(t). Since n(t) is a step function, it is not continuous at the points where the number of quarters needed changes. These are the points where the function jumps from one integer value to another.
5. Step 5: Conclude about the continuity of n(t). Since n(t) is not continuous at the points where it jumps, it is not continuous for all values in its domain. It is only continuous within the intervals between these jumps.