I Uniform convergence of difference quotient in higher dimensions

psie
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In ##\mathbb R## with ##f:[a,b]\to\mathbb R##, one can show that the difference quotient in the definition of the derivative of ##f## converges uniformly on ##[a,b]## to the derivative of ##f##. I wonder, is this true for ##\mathbb R^d## with ##d>1## too?
Let ##f:\mathbb R^d\to\mathbb R^d## and let ##f## be differentiable on an open subset ##U\subset\mathbb R^d##. We then have $$\lim_{h\to0}\frac{|f(h+a)-f(a)-Df(a)(h)|}{|h|}=0,\quad \forall a\in U.$$ ##Df(a)## is the linear transformation at ##a##, i.e. the derivative of ##f##, acting on ##h##. Let now ##K\subset U## be compact. Under what conditions on ##f## does the fraction above converge uniformly to ##0## on ##K##?

I'm partly reading this answer (and also my own textbook, where this claim seems to be made) and it seems to hold for ##d=1## with the help of the mean value theorem and the assumption that ##f## is continuously differentiable, but I have no clue how one would go about it for ##d>1##, since I don't think any mean value theorem/inequality holds for vector-valued functions of a vector variable. If you know a proof, I'd be happy to hear about it. :smile:
 
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psie said:
since I don't think any mean value theorem/inequality holds for vector-valued functions of a vector variable.
There is the mean value theorem for vector valued functions. See https://mathresearch.utsa.edu/wiki/index.php?title=Mean-Value_Theorems_for_Vector_Valued_Functions.

If you look at their proof you will see that you can, in similar way, define a new function ## F(t)=\textbf{a}\textbf{f}(\textbf{x}+t\textbf{J}) ## where ## \textbf{a}\in\mathbb{R}^m ## and ## \textbf{J}\in\mathbb{R}^n ## is a vector of ones. ## F’(t) ## will be ## \textbf{a}\textbf{f}’(\textbf{x}+t\textbf{J})\textbf{J} ##. By using $$ |\frac{F(t+h)-F(t)}{h}-F’(t)|\lt\epsilon $$ for ## t=0 ## you can get the next $$ |\frac{\textbf{a}\textbf{f}(\textbf{x}+h\textbf{J})-\textbf{a}\textbf{f}(\textbf{x})}{h}-\textbf{a}\textbf{f}'(\textbf{x})\textbf{J}|\lt\epsilon $$.
 
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