Answer:
If a one-to-one function is not continuous, it is sometimes increasing or decreasing. If a one-one function is continuous, it is always increasing or always decreasing.
For example, if
This function is one-to-one and it is not always increasing or decreasing.
Step-by-step explanation:
A statement that are one-to-one functions either always increasing or always decreasing is given.
It is required to explain the given condition.
To explain the given condition, consider a continuous and a not continuous function then explain about the one-to-one function.
If a one-to-one function is not continuous, it is sometimes increasing or decreasing.
For example, if
This function is one-to-one and it is not always increasing or decreasing.
If a one-one function is continuous, it is always increasing or always decreasing.
For example, if f(x)=x-3. It is a continuous function and it reaches a minimum value at some values of x and it then keeps increasing.
There will be the same output value for different input values. It does not pass the horizontal line test and so it is not a one-to-one function.