Taking the limit of a sequence of functions is a most natural and necessary process in analysis, and it is an important way of how new classes of functions arise. It is crucial to understand whether and when certain properties of the functions in the sequence (such as continuity or integrability) will pass to the limit function. The question is also equivalent to whether two different limit processes applied to a sequence of functions can be exchanged.
Suppose that \(\{ f_n(x)\}\) is a sequence of functions defined on a set \(E\text{,}\) and for any \(x\in E\text{,}\)\(f_n(x) \to f(x)\) as \(n\to \infty\text{.}\) Some of the questions can be raised for fairly general \(E\text{,}\) but one may first restrict to the case when \(E\) is an interval of \(\mathbb R\text{.}\) The main questions that concern us in this chapter are the following.
If each \(f_n(x)\) is continuous on \(E\text{,}\) does it imply that \(f(x)\) is continuous on \(E\text{?}\) If not, what kind of "bad" behavior can \(f(x)\) exhibit? What additional conditions on the convergence would guarantee that \(f(x)\) is continuous on \(E\text{?}\) Here we formulate the question using continuity over \(E\text{,}\) but we could also formulate the question using continuity at a specific point \(\bx_0\) in \(E\text{.}\) Then the answer to the question depends on whether we can exchange the following two limits to get an equality:
