One of the most important mathematical concepts in physics and pure mathematics is continuity. There’s a formal definition for it which for the moment isn’t too relevant, but for our purposes we can think of it in terms of smoothness. Put your finger at a point on the graph, and if the function smoothly approaches from all directions with no instantaneous jumps or gaps and it’s continuous at that point. But it can get kind of tricky. Here’s one that’s frequently presented in intro multivariable calculus classes (I first encountered it on page 904 of Stewart).

Its mug shot:

Ignore the rendering difficulties near the origin for the moment. The function is not technically continuous at the origin anyway, because the denominator is 0 where x and y are zero. But if we could find what value the function approaches we could make an appropriate redefinition and fill in the missing point. Let’s do it with some trickery. We can put our finger somewhere else on the graph and move it toward the origin and see what the value of the function is doing. And furthermore let’s do it from every direction at once! It’s easier than it sounds: just let y = mx and you’ve got every straight line approach at once for the infinite possible values of m. Thus:

Which is manifestly equal to 0 when x approaches 0. Case closed, let’s define f(0,0) = 0 and be done, with a function which is continuous everywhere.

Or is it? Every *straight line* approach to the origin is not the same as every approach. What about that parabolic-shaped ridge that seems to get pretty near the origin? Let’s approach the origin along that ridge by setting x = y^2.

Which is clearly not 0 at all! So we do have an instant jump at the origin, and thus the function is irretrievably discontinuous at that point.

Alas. But these things do happen, and it’s better to recognize them before they work their way into your theory undetected later and wreck everything.