REAL ANALYSIS

This is one exercise in Abbotts 2nd edition of understanding analysis. The section is titled Uniform convergence of a sequence of functions. Pay attention to the Theorem listed at the bottom.

Exercise 6.2.2. (a) Define a sequence of functions on R by

f n(x) ={ 1 if x = 1,1/2 ,1/3 ,...,1/n 0 otherwise

and let f be the pointwise limit of f n. Is each f n continuous at zero? Does f n f uniformly on R? Is f continuous at zero?

(b) Repeat this exercise using the sequence of functions

g n(x) = {x if x = 1, 1/2 , 1/3 ,..., 1/n. 0 otherwise.

(c) Repeat the exercise once more with the sequence

h n(x)={1 if x = 1/n x if x = 1, 1/2 ,1/3 ,..., 1/(n1) 0 otherwise

In each case, explain how the results are consistent with the content of the Continuous Limit Theorem (Theorem 6.2.6).

Theorem 6.2.6 (Continuous Limit Theorem). Let (f n) be a sequence of functions defined on AR that converges uniformly on A to a function f. If each f n is continuous at cA, then f is continuous at c.