Mean-value theorem for vector-valued functions

Matkowski, Janusz. "Mean-value theorem for vector-valued functions." Mathematica Bohemica 137.4 (2012): 415-423. <http://eudml.org/doc/247065>.

@article{Matkowski2012,
abstract = {For a differentiable function $\{\bf f\}\colon I\rightarrow \mathbb \{R\}^\{k\},$ 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 $M\colon I^\{2\}\rightarrow I$ such that\[ \{\bf f\}(x)-\{\bf f\}( y) =( x-y) \{\bf f\}^\{\prime \}( M(x,y)) ,\quad x,y\in I, \] are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.},
author = {Matkowski, Janusz},
journal = {Mathematica Bohemica},
keywords = {Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function; Lagrange mean-value theorem; Darboux property},
language = {eng},
number = {4},
pages = {415-423},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mean-value theorem for vector-valued functions},
url = {http://eudml.org/doc/247065},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Matkowski, Janusz
TI - Mean-value theorem for vector-valued functions
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 4
SP - 415
EP - 423
AB - For a differentiable function ${\bf f}\colon I\rightarrow \mathbb {R}^{k},$ 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 $M\colon I^{2}\rightarrow I$ such that\[ {\bf f}(x)-{\bf f}( y) =( x-y) {\bf f}^{\prime }( M(x,y)) ,\quad x,y\in I, \] are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.
LA - eng
KW - Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function; Lagrange mean-value theorem; Darboux property
UR - http://eudml.org/doc/247065
ER -