Continuous vs Broken Paths

Calculus 1 · Axiom Academy

Can you draw the whole function without lifting your pencil? When you can't, where it breaks tells you exactly how. The pencil test: one smooth motion, or a lift? A function is continuous when you can trace its whole graph without ever lifting your pencil — no breaks, no jumps, no holes. That one idea sits underneath almost everything in calculus: limits, derivatives, the guarantee that a moving thing actually passes through every value on its way. Watch a pen trace each curve. On the left it glides the whole way in one unbroken stroke — that's a continuous function. On the right the curve breaks, so the pen has to lift and set back down to keep going. That lift is a discontinuity. Continuous = the pen never leaves the paper. Discontinuous = at least one point forces a lift. Not all breaks are the same. Pick each type below and watch the curve redraw. Continuity needs one clean condition — — so compare the limit the curve is heading toward with the value actually plotted at that point, and see exactly which part fails. Each break fails the same test for a different reason: the value is missing, the two sides disagree, or no finite limit exists at all. A removable hole is the only break you can patch. The curve heads straight for the height 2 at , but nothing is plotted there. Drag the loose point up onto that limit value to plug the gap — the moment it lands at y = 2 , the pencil can pass straight through and the function is continuous.

This is the written version of the interactive lesson above. See the full Calculus 1 course.