fₙ(x) = xⁿ on [0,1]

Real Analysis · Axiom Academy

EXAMPLE Pointwise vs. Uniform Convergence of f_n(x) = x^n on [0,1] The textbook sequence that separates pointwise convergence from uniform convergence. Consider the sequence of continuous functions f_n(x) = x^n on the interval [0,1] . Find its pointwise limit, and determine whether the convergence is uniform on [0,1] . As n grows, each xⁿ stays near 1 close to x = 1 for longer, then drops to 0. The biggest gap from the limit (near x = 1) never shrinks — it stays at 1. You just worked through the classic example separating pointwise from uniform convergence. Pointwise limit: pointwise on [0,1] , where f(x) = 0 for and f(1) = 1 . Discontinuous limit: each f_n is continuous, but f has a jump at x = 1 . So it cannot be uniform: a uniform limit of continuous functions is continuous — here the limit is not, so the convergence is not uniform. Confirmed directly: for every n (approached as ), which does not go to 0 . The principle: uniform convergence preserves continuity; pointwise convergence does not. This is exactly why uniform convergence matters: it is the condition that lets you swap a limit with continuity, integration, or differentiation.

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