TITLE: Good example of a non-continuous function all of whose partial derivatives exist
QUESTION [7 upvotes]: What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?

REPLY [3 votes]: I was travelling when this came up.
With the same results as Georges Elencwajg's comment and Marco's recent answer, take
$$ f(x,y) = \frac{2 x^2 y}{x^4 + y^2} $$ and set to $0$ at the origin $(0,0).$  Along any line  through the origin $ x = a t, \; y = b t$ the limit is 0, as 
$$ | f(a t, b t) | \leq a^2 | t / b | .  $$ However, along  the parabola $y = x^2$ the value is 1, and along the parabola $y = - x^2$ the value is $-1.$
To get "directional derivative" 0 in every direction through the origin switch to
$$ g(x,y) = \frac{2 x^3 y}{x^6 + y^2} $$ as
$$ | g(a t, b t) | \; \leq \; t^2 \; | a^3 / b |   $$ when $b \neq 0,$ but then if $y = \pm x^3$...the directional derivative  generalizes to the Gateaux derivative in other settings.