course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
17 |
|
3.II.11A |
2001 |
Prove that if all the partial derivatives of $f: \mathbb{R}^{p} \rightarrow \mathbb{R}$ (with $p \geqslant 2$ ) exist in an open set containing $(0,0, \ldots, 0)$ and are continuous at this point, then $f$ is differentiable at $(0,0, \ldots, 0)$.
Let
$$g(x)= \begin{cases}x^{2} \sin (1 / x), & x \neq 0 \ 0, & x=0\end{cases}$$
and
$$f(x, y)=g(x)+g(y) .$$
At which points of the plane is the partial derivative $f_{x}$ continuous?
At which points is the function $f(x, y)$ differentiable? Justify your answers.