Functions Approaching Functions

Real Analysis · Axiom Academy

Functions Approaching Functions A sequence of numbers can approach a limit. So can a sequence of functions — but there are two very different ways an entire graph can "get close." What does it mean for a whole function to approach another? A sequence of numbers like marches to a single limit. But a sequence of functions is a sequence of whole graphs. For the graphs to "approach" a limit function f , every height has to settle down. The catch: there are two genuinely different ways that can happen — and the difference decides whether limits, continuity, and integrals survive the process. Watch the classic example. Each f_n(x) = x^n on [0,1] is a perfectly smooth, continuous curve. Press play and let n climb: every point left of 1 gets dragged down toward 0 , while the right end stays nailed at the height 1 . The limit isn't smooth at all — it tears into a flat 0 with a lone point sitting up at (1,1) . The "sup gap" is the single worst vertical distance between f_n and the limit f anywhere on [0,1] — and here it never shrinks below 1 . Uniform: the whole graph closes in at once Here is a friendlier sequence: on [0,1] , heading for the limit f(x) = x . Drag n up and watch the entire copy slide down toward the line as one rigid piece. The worst gap anywhere — the sup gap — is exactly , and it marches to 0 . When the single worst distance goes to 0 , the convergence is uniform .

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