$f_n : \Bbb R → \Bbb R$ : $f_n (x) = \frac{nx}{1+|nx|}$
a) To show that the function sequences f are continuous for all n ∈ ℕ.
b) Determine the limit function
c) Why the function sequence does not converge uniformly
My proposed solution:
a) Since for all n $f_n$ is the quotient of 2 continuous functions and the denominator never becomes 0, fn is continuous for all n, actually this is the easiest part.
b) I calculated the limit and got 1 for x>0 and -1 for x<o.
c) Can we say that because f(x) from part b) is not continuous --> fn is not uniformly convergent ?
a) $f_n$ is continuous for all $n\in\mathbb{N}$ because it is the quotient of continuous functions whose the denominator is always positive
The answer you gave in b) it is correct and so it is the answer to c). If $f_n$ converged uniformily then the limit function would need to be continuous. More info