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Let $f:[a,b] \rightarrow \mathbb{R}$ be a function which is continuous on $(a,b]$ and differentiable on $(a,b)$. Is there any function such that $f(b)-f(a)≠(b-a)f'(x), \forall x\in (a,b)$?

There was a typo, and now it's edited. I wanted to know whether compactness of a set where $f$ is continuous on is essential.

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    Also, if your current version $f(b)-f(a)\ne (b-a)f'(x),\forall x\in (a,b)$ is correct, then any nonlinear function satisfies your criteria. I think you meant $\exists x\in (a,b)$.2012-11-14

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Let $f(0)=-20, f(x)=0 \text{ for } x \in (0,1]$. Is this the sort of example you were thinking of?