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Is it true that a discontinuous function cannot have both an absolute maximum and an absolute...

Question:

Is it true that a discontinuous function cannot have both an absolute maximum and an absolute minimum value on a closed interval?

(Give reasons for your answer.)

Absolute Maximum and Absolute Minimum:

A continuous function over a closed and bounded interval will always reach its absolute maximum and minimum, if the function is not continuous these values may exist or not.

Answer and Explanation: 1

It is false, for example the following function:

{eq}f\left( x \right) = \left\{ \begin{array}{l} - 1\quad - 3 \le x \le 0\\ 1\quad \;\;\;\;0 < x \le 3 \end{array} \right. {/eq}

is not continuous at the interval {eq}[-3,3] {/eq}, because it is not continuous at the origin but have both, the absolute maximum (1) and the absolute minimum (-1) at this interval.


Learn more about this topic:

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Continuous Functions Theorems

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Chapter 5 / Lesson 10
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Continuous functions are functions that have their conditions satisfied between multiple points, appearing as an uninterrupted line when graphed. See examples of how this is represented in the intermediate value theorem and the extreme value theorem.


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