Mathematics
Limits and Continuity at a Point in Topological Spaces
Quick fact
In a topological space, a function is continuous at a point if every open set containing the function value contains the image of some open set containing the point—no distances, no epsilon-delta, yet it exactly captures the idea of 'no sudden jumps'.
Why this is interesting
You already know that a continuous function has no jumps—but what if we remove the ruler that measures distance? How can we still define limits and continuity?