linguistic:translation, structural:scholar_edition
Pointwise and uniform limits

Given a sequence f 1, f 2, …: I → R {\displaystyle f_{1},f_{2},\dotsc:I\to \mathbb {R} } of functions such that the limit f (x):= lim n → ∞ f n (x) {\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)} exists for all x ∈ D, {\displaystyle x\in D,}, the resulting function f (x) {\displaystyle f(x)} is referred to as the pointwise limit of the sequence of functions (f n) n ∈ N. {\displaystyle \left(f_{n}\right)_{n\in N}.} The pointwise limit function need not be continuous, even if all functions f n {\displaystyle f_{n}} are continuous, as the animation at the right shows. However, f is continuous if all functions f n {\displaystyle f_{n}} are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.