Uniform convergence

Let E be a set and Y be a metric space. Consider functions fn : E → Y for n = 1, 2, . . . . We say that the sequence (fn) converges pointwise on E if there is a function f : E → Y such that fn(p) → f(p) for every p ∈ E. Clearly, such a function f is unique and it is called the pointwise limit of (fn) on E. We then write fn → f on E. For simplicity, we shall assume Y = R with the usual metric. Let fn → f on E. We ask the following questions: (i) If each fn is bounded on E, must f be bounded on E? If so, must supp∈E |fn(p)| → supp∈E |f(p)|? (ii) If E is a metric space and each fn is continuous on E, must f be continuous on E? (iii) If E is an interval in R and each fn is differentiable on E, must f be differentiable on E? If so, must f ′ n → f ′ on E? (iv) If E = [a, b] and each fn is Riemann integrable on E, must f be Riemann integrable on E? If so, must ∫ b a fn(x)dx → ∫ b a f(x)dx?