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.

>>10
Oh, oops. I read the OP's request as "continuous at 0 on L", not continuous on the entirety of L. I think it can be rectified by redefining the function somewhat;

f(x, y) = 1 if y = x^2 and x != 0
        = 0 if y < x^2/2 or y > 3x^2/2 and x != 0
        = 0 if x = 0
        = 2y/x^2 - 1 if y >= x^2/2 and y < x^2
        = -2y/x^2 + 3 if y > x^2 and y <= 3x^2/2.

basically the same as before except it slopes up to the parabola. I know there's some rational function that has the property the OP is looking for that would be a lot nicer, but I can't fucking remember it.