Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f : \mathbb R^n \rightarrow \mathbb R^m$ and $a \in \mathbb R^n$such that $\forall i \in [1,n], \large \frac{\partial f}{\partial x_i}(a)$ exists.

Is $f$ Frechet differentiable ?

I'd say no, but I don't know any counter-examples...

share|improve this question
    
The answer becomes positive if the partial derivatives exist and are continuous functions. –  Siminore May 11 at 9:53

1 Answer 1

up vote 4 down vote accepted

Let $n = 2$, and $m = 1$, and consider

$$f(x) = \begin{cases} \frac{xy}{x^2+y^2} &, (x,y) \neq (0,0)\\ \quad 0 &,(x,y) = (0,0).\end{cases}$$

Then $f$ isn't even continuous at $(0,0)$, but the partial derivatives exist. (The partial derivatives exist on all of $\mathbb{R}^2$. $f$ is Fréchet-differentiable everywhere except at $(0,0)$.)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.