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.
There's an intersection between the parabola and every line through the origin, except the trivial x-axis line. Therefore every line through the origin has one point where f(x,y)=1 and it is not continour at that point.