Mathematics · Advanced university
Pointwise versus uniform convergence · revision notes
The cutoff N may depend on both ε and the point x.
Key ideas and conditions
Pointwise
The cutoff N may depend on both ε and the point x.
Uniform
One N must work for all points in the domain at once.
Supremum norm
Uniform convergence is equivalent to sup_x abs(f_n(x)−f(x)) tending to zero.
Continuity
A uniform limit of continuous functions is continuous; pointwise limits need not be.
Explain without looking
- For a fixed 0<a<1, show x^n converges uniformly to zero on [0,a].
- State why interchanging a limit and an integral on a finite interval is justified for a uniformly convergent sequence of continuous functions.
Further reading and review references
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