Pointwise Convergence

A sequence of functions f n : X → Y : subscript 𝑓 𝑛 → 𝑋 𝑌 f_{n}:X\to Y into a topologica space Y 𝑌 Y is said to pointwise converge to a function f : X → Y : 𝑓 → 𝑋 𝑌 f:X\to Y when:

∀x:X.limn→∞fn(x)=f(x)\forall x:X.\,\lim_{n\to\infty}f_{n}(x)=f(x)

Intuitively, this means that f n ⁢ ( x ) subscript 𝑓 𝑛 𝑥 f_{n}(x) can converge to f ⁢ ( x ) 𝑓 𝑥 f(x) at different rates for each x : X : 𝑥 𝑋 x:X .