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
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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.
More to read
5.1 Sequences
A sequence of functions gives you one ordinary sequence at every single point. Fix x equal 0.3 and watch the values f_1(0.3), f_2(0.3), and so on. Then move to another x and watch a different sequence. Pointwise convergence means every one of these frozen point sequences settles down to a limit.
The catch is that each point may settle at its own speed. One x value might be close to its limit by n equal 10 while another needs n equal 10,000.
2.4 Continuity
Continuous functions can surprise you when you take limits of whole sequences of them. The classic example is x to the power n on the interval from 0 to 1. Every curve in the sequence is smooth and unbroken, yet the pointwise limit jumps at x equal 1.
This is exactly why analysts invented a stronger kind of convergence. Continuity is about nearby points staying close in value. To preserve it through a limit, you need control that works across all points at once, not point by point.
6.1 Power Series and Functions
Power series are the success story of uniform convergence. Inside their disc of convergence they behave beautifully, and on any smaller closed disc the convergence is uniform. That uniformity is what lets you treat an infinite sum almost like a long polynomial.
Compare this with a raw sequence of functions, where each point converges on its own schedule. Power series converge as a team on compact subsets.
6.2 Properties of Power Series
Term by term differentiation and integration of power series work because the convergence is uniform on closed subintervals. You differentiate each term, and the new series still converges to the derivative of the sum. Nothing like this holds for pointwise convergence in general.
So when you see a theorem that swaps a limit with an integral or a derivative, look for uniformity hiding in the hypotheses.
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Where it sits
17 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.