Let be a function such that along every line and the line , the function is continuous at . By this I mean if we let be one of the lines in the x-y plane given by rotating an angle anti-clockwise from the x-axis, and let be a sequence converging to s.t. for all , then as .

Is it true then that is continuous at ? So if we take an arbitrary sequence converging to , say , then does ? I was thinking that the each element lies on some line so that there are countably many lines that the sequence lies on, but I'm not sure if that means anything. Is the statement even true?

Thanks for any help, just something that was bothering me