Directional Derivative and differentiability

Question

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 $a$?

Vincent Boelens · Accepted Answer · 2014-04-12T18:15:31.763

2

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 $$\lim_{h\to 0}\frac{f(a+h)-f(a)-Df(a)h}{\|h\|}=\lim_{h\to 0}\frac{f(a+\|h\|v)-f(a)}{\|h\|}-Df(a)\left(\frac{h}{\|h\|}\right)= \lim_{h\to 0}\frac{f(a+\|h\|v)-f(a)}{\|h\|}-Df(a)v=0$$

edited Apr 12 '14 at 18:15

answered Apr 12 '14 at 14:53

Vincent Boelens

3,814

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 '14 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 '14 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 '14 at 16:53
@Anoldmaninthesea. Not sure if I can help you with a reference. Most the books I know are in German. You could try Spivak's Calculus on Manifolds. – Vincent Boelens Apr 12 '14 at 17:18
@Anoldmaninthesea. Uniform convergence doesn't guarantee continuity of the partial derivatives. It makes sure that $\lim_{h\to 0}\frac{f(a+h)-f(a)}{|h|}-Df(a)h=0$, where $Df(a)$ is the Jacobian. – Vincent Boelens Apr 12 '14 at 17:22
How? I'm sorry I'm not seeing. Could you help me? – An old man in the sea. Apr 12 '14 at 17:50
@Anoldmaninthesea. I provided a proof – Vincent Boelens Apr 12 '14 at 18:15
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@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 '14 at 18:20
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@Anoldmaninthesea. ...Introduction to mathematical analysis by Kriz, Differential forms by Weintraub and Multivariable analysis by Satish. – Mark Fantini Apr 12 '14 at 18:22
@Fantini, Obrigado. ;) – An old man in the sea. Apr 12 '14 at 18:39
@VincentBoelens, I understood your comment, but what I didn't understood was the relation with uniform convergence. Now I think I get it. With unif. conv, I'm sure that lim goes to zero since $sup_{||h||\leq 1} |\frac{f(a+h)-f(a)}{||h||}-Df(a)h| \rightarrow 0$, as h goes to zero. Many Thanks, Vincent Boelens – An old man in the sea. Apr 12 '14 at 18:54

Directional Derivative and differentiability

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