Epsilon-Delta Definition of Continuity

Real Analysis · Axiom Academy

LESSON The Epsilon-Delta Definition of Continuity "No breaks in the graph," made exact: for every tolerance you demand on the output, an input range delivers it. The output value we aim at is f(c) . Pick a tolerance — a horizontal band on the y -axis. To be continuous at c , the function must let you pick a radius — an interval on the x -axis — so small that every input from it lands inside the band. Read it left to right: for every , there exists a , that does the job. 2. Tighter Tolerance, Narrower Range Here is the engine of the definition: depends on . Demand a tighter output band, and you must answer with a narrower input interval. The animation shrinks in steps; for each, it draws the widest that still keeps the curve inside the band. Watch the two contract together. A roomy band. A wide -interval already keeps every f(x) inside — the response is easy. A thin band. Only a tight -interval keeps f(x) inside — the response must shrink to match. Where f rises fast, a small horizontal step causes a big vertical jump, so has to be smaller for the same . No matter how small gets, a working still exists. That "always" is exactly continuity. Worked response: f(x)=2x at c=3 Here f(c)=6 . For any , if then . To force , take . So you can always answer — and the steeper the line, the smaller that .

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