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.

>>4
Correct. A rather nice one is

f(x,y) = q when y = (p/q)x, x != 0
f(x,y) = 0 where y = rx, r irrational, x != 0
f(0,0) = 0