- #1

Lancelot1

- 28

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Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ?

How do you prove such a thing ?

Thank you !

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- MHB
- Thread starter Lancelot1
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In summary, a monotonic function is a mathematical function that either always increases or always decreases over a given interval. An increasing monotonic function is a function that always increases over a given interval. A closed interval is a set of real numbers between two specified values, where both endpoints are included in the set. An increasing monotonic function cannot be discontinuous in a closed interval because it must be defined for all values in the interval. Continuity and monotonicity are both properties of functions, with continuity ensuring no breaks or gaps in the graph and monotonicity indicating a constant increase or decrease. In a closed interval, an increasing monotonic function must also be continuous because its graph has a positive slope and no breaks or gaps.

- #1

Lancelot1

- 28

- 0

Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ?

How do you prove such a thing ?

Thank you !

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- #2

HOI

- 921

- 2

For example the function f(x)= x for \(\displaystyle 0\le x\le 1\), f(x)= x+1 for \(\displaystyle 1\le x\le 2\), and, in general, f(x)= x+ n for \(\displaystyle n\le x\le n+1\) for n an integer is monotone increasing for all x but is discontinuous at every integer.

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.

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