Recently in my advanced calculus class we discussed how the uniform limit of a sequence of continuous functions is itself continuous. One of my students turned the question around: If the limit of a pointwise sequence of continuous functions is continuous, does that mean that the convergence is uniform?
The answer is “No,” although I couldn’t think of a counterexample off the top of my head. I found one (not surprisingly) on Math.SE. Although there are some good discussion and multiple good answers to this question given there, I’m going to discuss Ilmari Karonen’s answer.
Let be given by
Each is continuous (piecewise, but that’s fine). For any , there exists N such that for all . Since for all n, this means that for all . The zero function is clearly continuous.
However, for each we have that . This means that when there cannot be an such that for all and . Thus the convergence cannot be uniform.