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Mean-value theorem for vector-valued functions. (English) Zbl 1274.26009
Summary: For a differentiable function $$f\: I\rightarrow \mathbb {R}^k$$, where $$I$$ is a real interval and $$k\in \mathbb {N}$$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $$M\: I^2\rightarrow I$$ such that $f(x)-f(y)=(x-y)f'(M(x,y)),\; x,y\in I,$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 26E60 Means
##### Keywords:
Lagrange mean-value theorem; Darboux property
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