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My question is similar, but not equal to this...Question on linearity of directional derivative

Let $f'_{h}(a)$ be the directional derivative. And for the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$

$f'_{h}(a)=\sum^n_{i=1}\frac{ \partial f}{\partial x_i}(a) \cdot h_i$

Is f differentiable at $a$?

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1 Answer 1

up vote 2 down vote accepted

No, it isn't. The linked question gives a counterexample. But it is true if $$\lim_{t\to 0} \frac{f(a+th)-f(a)}{t}$$

converges uniformly for all $h$ in the unit ball.

Edit: I'll include a proof.

Put $v:=\frac{h}{\|h\|}$. Then we have $$\lim_{h\to 0}\frac{f(a+h)-f(a)-Df(a)h}{\|h\|}=\lim_{h\to 0}\frac{f(a+\|h\|v)-f(a)}{\|h\|}-Df(a)\left(\frac{h}{\|h\|}\right)= \lim_{h\to 0}\frac{f(a+\|h\|v)-f(a)}{\|h\|}-Df(a)v=0$$

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I hope you don't mind me asking, but do you know of a book that explains well multivariate analysis? Which one do you use. The one I use leaves many things hidden. A book that gives intuition, and still remains some formalism would be great. –  An old man in the sea. Apr 12 at 16:26
    
And if it converges uniformly, then the directional derivative is continuous, which in this exercise, it will make the partial derivatives continuous. Hence the function would be C^1 which implies differentiability. I am correct? –  An old man in the sea. Apr 12 at 16:51
    
But how can I be sure that f is continuous on a neighbourhood of point a? –  An old man in the sea. Apr 12 at 16:53
1  
@Anoldmaninthesea. Book recommendations for multivariable analysis: Multidimensional real analysis by Duistermaat, volumes one and two, Analysis in vector spaces, by Mustafa... –  Mark Fantini Apr 12 at 18:20
1  
@Anoldmaninthesea. ...Introduction to mathematical analysis by Kriz, Differential forms by Weintraub and Multivariable analysis by Satish. –  Mark Fantini Apr 12 at 18:22

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