# Values of functions on a closed interval
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slide: https://hackmd.io/@ccornwell/extreme-values
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<h3>Think of a function...</h3>
- <font size=+1>so that on the interval $(1,5)$, the function does not have a maximum value.</font>
- <font size=+1>Did something _need_ to happen (at some point) in your interval for the function to have no maximum? Or would many (most?) functions not have a maximum on this interval?</font>
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<h3>Think of a function...</h3>
- <font size=+1>so that on the interval $[1,5]$, the function does not have a maximum value.</font>
- <font size=+1>Did something _need_ to happen (at some point) in your interval for the function to have no maximum?</font>
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<h3>The Extreme Value Theorem</h3>
- <font size=+2>If $f(x)$ is continuous at every point in a closed interval $[a,b]$, then it has both an absolute maximum value, and absolute minimum value, in $[a,b]$.</font>
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