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.

>>12
Your example in 5 has issues. For instance, on the line y = x. f(x,y) = 1 for x != 0 because y = 1/1 * x. But f(0,0) = 0, so it isn't continuous on the entirety of the line.