The definition of a limit, made into a game. You pick a tolerance ε — a horizontal band of width 2ε around the target height L. The machine tries to respond with a δ — a window of width 2δ on x around the point c — narrow enough that every input inside it lands inside your band. Shrink ε and watch δ chase it down. When the machine can always answer, the function is continuous at c. When it can't, you've caught a discontinuity red-handed.