Understanding a proof of inexistence of max nor min

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In summary: All clear, just a typo. In summary, a function with no maximum or minimum at an endpoint can still be continuous and differentiable, as proven by examples using the conditions of a function not having a local maximum or minimum value. The proof may vary, but the concept remains the same.
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mcastillo356
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I've got a proof of the inexistence of a local maximum value nor a local minimum value at the origin of coordinates for a certain function, and need advice to understand it
Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value
A function with no max or min at an endpoint. Let

Show that is continuous on and differentiable on but it has neither a local maximum nor a local minimum value at the endpoint
I've already proven it is continuous on and differentiable on , and here is the proof that it has not loc min or loc max:
Background
A function has not a maximum local value at in its domain if for all can always be found , such that and
A function has not a minimum local value at in its domain if for all can always be found , such that and


Proof I don't understand:

For all we have and , therefore .
Let .Exists one such that . Then for all we have:
. Then and
No matter how small is we have . And is proven that cannot be maximum nor minimum.
 
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There could be more specific detail in the end of the proof. Like:

What part do you not understand?
 
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I have actually proven it, based on the same basis. I know eventually , but in a Spanish math forum I stumbled across this algebra. This proof is smarter than mine, shorter, direct, but incomprehensible

For all we have and , therefore .
Right
Let .Exists one such that . Then for all we have:
. Then and
This quote is imposible for me to understand. How does it fit the conditions of
mcastillo356 said:
A function has not a maximum local value at in its domain if for all can always be found , such that and
A function has not a minimum local value at in its domain if for all can always be found , such that and

No matter how small is we have . And is proven that cannot be maximum nor minimum.
Right
 
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From post #1:
mcastillo356 said:
A function has not a maximum local value at in its domain if for all can always be found , such that and
mcastillo356 said:
Right
Let .Exists one such that . Then for all we have:
. Then and
This inequality seems to have a typo in it: , and likely should be . The inequality as stated is certainly true, but makes the subsequent work harder to understand.
 
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Thanks, PF!
 

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