I got stuck on two problems that involves finding the pointwisse limits of piecewise functions. Hope someone can give me a hand.
1) Let be a sequence that contains each rational in [0,1] precisely once. For each , define
if x is irrational
Prove that converges pointwise on [0,1] to a function that is not R-integrable.
I guess that this sequence of functions converges to the rational characteristic function, but I can't really show as n approaches infinity.
I tried this again and I got this far: if x is irrational and if x is rational.
Consider: absolute value of = if x is irrational. Choose
If x is rational, . I think f(x)=1 because eventually as n approaches infinity.
2) For define by if and if . Prove that does not converge uniformly on [0,1] but converge uniformly on [M,1] where 0<M<1.
I don't know what the pointwise limit of , so I can't go any anywhere. It would be really helpful if someone helps me how to find the pointwise limit