(1.) State whether the following functions are continuous or not for all values in their domain.
Function 1: The number $f(t)$ of fish in a lake at day t of the fishing season.
Function 2: The length $L(t)$ of a fish t years after it is hatched.
General Rule of Thumb: Bring it to Statistics: Classifying Quantitative Variables:
If you can count it, it is discrete.
If you can measure it, it is continuous.
We can count (not measure) the number of fish in a lake.
This implies that the number of fish in a lake is a discrete variable.
We can measure (not count) the length of a fish after it is hatched.
This means that the length of a fish is a continuous variable.
Specific Rule of Thumb: Bring it to Calculus: Stating Whether Functions are Continuous or Not
If a function can take only distinct, separate values, it is discrete.
A function is discrete if the range consists of distinct, separate values.
The range of the function are only whole numbers.
We cannot have decimal values because we cannot have 3.7 fish.
The number of fish can be 3 fish.
The number of fish cannot be 3.7 fish.
Hence, Function 1 is discrete.
Function 1 is not continuous.
If a function can take any value within an interval, it is continuous.
A function is continuous if the range includes every value in at least one interval.
The length of fish can be 3 cm.
The length of fish can also be 3.7 cm.
The length of fish can also be 3.754789 cm.
The length of fish can also be 3.754789 ... cm. continuing
Hence, Function 2 is continuous.