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I have a question concerning the Mean Value Theorem (and maybe Rolle's Theorem).

In my calc book by Stewart, the concept of both theorems seemed to be thrown out of nowhere with a bunch of conditions and statements like.

"If $f$ is differentiable on an open interval $(a,b)$, then..."

So here is what i don't understand, why can't the interval be closed for differentiability? And what was the motivation behind creating the two theorems? I don't see how anyone, one day, could sit down and just write down a bunch of rules and conclude a formula and give a name to it.

I won't take the equation as face value and accept it like the rest of the mindless sheeps in my university

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    What does "differentiability at a point" mean?2012-12-19
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    It means f'(at a point) exists or in some books, they say the function is approximately *linear* near that point2012-12-19
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    The function can certainly be differentiable on the closed interval, and more. But the point is that the theorem even holds when we don't have differentiability at an end point. That makes the theorem applicable in a wider variety of situations. For example, the commonly occurring function $f(x)=\sqrt{x}$ is not differentiable at $0$, but the Mean Value Theorem can be used on this function to conclude that there is a $c$ between $0$ and $x$ such that $f'(c)=\frac{\sqrt{x}-0}{x-0}$.2012-12-19
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    Maybe because I am still inexperienced, but doesn't it stating on an open interval seem to eliminate the possibility of being differentiable on a closed interval? Why doesn't it state **or**2012-12-19
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    Mine was a rhetorical comment... meant to lead him to the crux of what he's asking about.2012-12-19
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    @sizz: When we say differentiable on an open interval, we do not **commit** ourselves to what might be happening beyond, so we are saying **less** than when we say differentiable on a closed interval.2012-12-19

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