Drawing Without Lifting

Real Analysis · Axiom Academy

Some graphs you can trace in a single stroke. Some force you to lift your pen — and that lift is exactly what we mean by a discontinuity. The whole idea, in one pen stroke Before any formula, continuity has a picture you already trust: a graph is continuous if you can draw it without lifting your pen off the paper. No breaks, no gaps, no sudden jumps — your hand never leaves the page from one end to the other. Watch the pen trace this curve from left to right. It glides over every point in between, drawing the graph behind it in one unbroken stroke — the pen never lifts. That is what "continuous" looks like. One continuous curve, one continuous stroke — the pen stays on the page the whole way across. Where the pen is forced to lift Here is a different graph. It sits at one height to the left of x = 0, then suddenly sits at a higher one to the right. Slide the pen across and watch what happens at x = 0: there is no way to get from the lower piece to the higher one without lifting off the page. That break is a jump discontinuity. One jump → exactly one forced lift. A graph with a jump cannot be drawn in a single stroke — so it is discontinuous there. Continuous, or not? Trace each one Three more graphs. Pick one, then drag the pen all the way across it. Can you predict which need a lift before you try? Watch out — a sharp corner is perfectly fine to trace, but a single missing point is not.

This is the written version of the interactive lesson above. See the full Real Analysis course.