Please explain how the results are consistent with the conte

Please explain how the results are consistent with the content of the Continuous Limit Theorem.

Solution

;1) Clearly the (pointwise) limit function f(x) is 1 at all x=1,1/2,1/3....as given any n , f_k (1/n) =1 for all k >=n.

and f(x) = 0 otherwise. Convergence is not uniform (as the choice of M depends on the value of x)

Each f_n is continuous at x=0 as it is the constant function 0 in a neighbourhood of 0 not containing 1/n

f(x) is not continuous at x =0 as any neighbourhood of 0 contains a point of the form 1/m where the function value is 1. In other words ,So |f(0)-f(1/n)| =1 and cant be made smaller.

As the convergence is not uniform, the limit function f(x) is not guaranteed to be continuous.