Editing Continuous function (section)

====Pointwise and uniform limits====
[[File:Uniform continuity animation.gif|A sequence of continuous functions <math>f_n(x)</math> whose (pointwise) limit function <math>f(x)</math> is discontinuous. The convergence is not uniform.|right|thumb]]
Given a [[sequence (mathematics)|sequence]]
<math display="block">f_1, f_2, \dotsc : I \to \R</math>
of functions such that the limit
<math display="block">f(x) := \lim_{n \to \infty} f_n(x)</math>
exists for all <math>x \in D,</math>, the resulting function <math>f(x)</math> is referred to as the [[Pointwise convergence|pointwise limit]] of the sequence of functions  <math>\left(f_n\right)_{n \in N}.</math> The pointwise limit function need not be continuous, even if all functions <math>f_n</math> are continuous, as the animation at the right shows. However, ''f'' is continuous if all functions <math>f_n</math> are continuous and the sequence [[Uniform convergence|converges uniformly]], by the [[uniform convergence theorem]]. This theorem can be used to show that the [[exponential function]]s, [[logarithm]]s, [[square root]] function, and [[trigonometric function]]s are continuous.