I'm reading a book which gives this theorem without proof:
If a and b are any two points in an interval on which ƒ is differentiable, then ƒ'
takes on every value between ƒ'(a) and ƒ'(b).
As far as I can say, the theorem means that the fact ƒ' is the derivative of another function ƒ on [a, b] implies that ƒ' is continuous on [a, b].
Is my understanding correct? Is there a name of this theorem that I can use to find a proof of it?
Answer
The result is commonly known as Darboux’s theorem, and the Wikipedia article includes a proof.
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