An increasing monotonic function is a mathematical function where the output values increase as the input values increase. This means that as the input increases, the output will always increase or stay the same. It is also known as a non-decreasing function.
A closed interval is a set of real numbers between two endpoints, where both endpoints are included in the set. For example, the closed interval [0, 1] includes all real numbers between 0 and 1, including 0 and 1.
No, an increasing monotonic function is not always continuous. It is possible for a function to be increasing but not continuous, meaning that there are points where the function has a jump or discontinuity.
Yes, it is possible for an increasing monotonic function to be discontinuous in a closed interval. This means that there is at least one point within the closed interval where the function has a jump or discontinuity.
To prove that an increasing monotonic function in a closed interval is continuous, we can use the Intermediate Value Theorem. This theorem states that if a function is continuous on a closed interval and takes on two values at the endpoints, then it must also take on every value in between. Therefore, if we can show that the function takes on the same value at the endpoints, we can conclude that it is continuous in the closed interval.