Monsieur Ejemplé

Give an example of a function f:R^2 --> R s.t. the function is not continuous at zero, but for any (straight) line L passing through the origin, the induced function f|L is continuous on L.

f(x,y) = 1 when y = x^2 and x != 0
f(x,y) = 0 everywhere else.

There are more nicely defined functions with the same property, but I'm too lazy to come up with one right now.