Limits and Continuity

Intro to Proofs · Axiom Academy

LESSON The Epsilon–Delta Definition Making "arbitrarily close" precise: for every tolerance on the output, a window on the input must keep the function inside. Take a function f near an input a . We say its limit there is L when output-closeness can always be forced by input-closeness. The output target sets a horizontal band around L ; we must answer with an input window around a whose graph stays inside that band. The two quantifiers — "for every " then "there exists a " — read naturally as a two-player game . The order is everything: the Challenger moves first, so your is allowed to depend on the you were handed. Names any tolerance , however tiny, demanding f(x) land within of L . Their job is to make you fail. Replies with a window . If every x within of a keeps f(x) within of L , the round is won. Let's actually win the game for . The trick is to solve backward : start from the output condition we need and unwind it into an input condition. The slope of 3 does the work. Reading the last line: forcing |x - 2| below is exactly what forces |f(x) - 5| below . So the choice writes itself. Continuity is the limit definition with one extra demand: the limit must equal the function's actual value there. No hole, no jump — the curve passes through the point it approaches. In – terms: for every there is a with . We drop the 0 < from 0 < |x-a| because here the point x = a is allowed in — the function must agree with its own limit.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.