Definition 8.1.5: Pointwise Convergence

A sequence of functions { f_n(x) } with domain D converges pointwise if for each fixed x₀ D in the domain the numeric sequence { f_n(x₀) } converges. In other words: for each fixed x₀ and any given > 0 there exists a positive integer N such that

| f_n(x₀) - L | < whenever n N

for some limit L. Note that the limit L depends on x₀, while the integer N depends on x₀ and .

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