I know its a pretty elementary question, but I never felt like I've had any sort of reasonable explanation of why. As I understand, we can define continuity for a function f: ℝn→ℝ as: For any ε>0 there exists a δ>0 such that for all x st 0< lx - al < δ then lf(x) - f(a)l < ε Alright, so in other words, we can always define a "disc" in ℝn such that all function values belonging to the points in that disc are less that our chosen epsilon. So let's say we find that every linear path that passes through a is continuous at the point a. Let's also say that for each linear path, there is a maximum delta, call it δ', such that for all linear paths, for every ε we can choose δ' such that for all x 0< lx - al < δ' → lf(x) - f(a)l < ε . Doesn't this prove the continuity? I know there are counter examples that will show that if you take some other path, you may get a different limit. But it still seems to me that that the above should be true.