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Consider a function $$ f:\mathbb{R}^n\rightarrow\mathbb{R}^m $$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$. Does there ...

I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that \begin{equation} f(x)-f(x')\approx (x-x')^...