Pointwise and Uniform Convergence
A sequence of functions can converge at every point and still lose continuity in the limit. Uniform convergence is the stronger condition that preserves continuity and permits term-by-term integration.
What a learner can do afterwards
- Give a sequence of continuous functions with a discontinuous pointwise limit
- State the difference between the two definitions in terms of quantifier order
- Say what uniform convergence licenses that pointwise convergence does not
1 · Read
A sequence of functions gives one ordinary sequence at every single point. Pointwise convergence means each frozen point sequence settles, possibly at its own speed. For x to the n on [0, 1], every point below 1 tends to 0 while 1 stays at 1. The pointwise limit is 0 before 1 and 1 at 1: a jump built from smooth curves.
Uniform convergence means one N works for every x at once: past N, the whole graph stays within epsilon of the limit. The difference is quantifier order. Pointwise picks x before N, so N may use x. Uniform picks N before x, so it cannot. A student claiming pointwise gives one N for all x has the order backward.
Uniform limits of continuous functions stay continuous, which pointwise cannot promise. The reverse fails: a continuous limit never proves uniformity. Uniformity also licenses term by term integration of series over [a, b]. Power series earn this treatment on closed subintervals, which is why they differentiate and integrate term by term.
Pointwise lets each point keep its own schedule, uniform makes the whole graph arrive together, and only the team arrival preserves continuity and the integral.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.