Pointwise convergence of piecewise functions

[Nemesis10192]

Hi guys, so I have just been doing some problems on point-wise convergence of functions which has been going fine. However, suddenly the questions on my practice sheets have changed in style to defining f_n(x) piecewise with the pieces dependant on a relation between x and n:

i.e

f_n(x)= 0 if x<=n
x-n if x> n.

This is really confusing me...I have no problems using the function by itself...I.e f_2(x)= 0 if x<=2 and x-2 if x>2. But I'm getting really confused about how to workout the pointwise limit...I mean as n-> infinity given a particular x it will always eventually be less than n, so is the pointwise limit just f(x)=0 for all x in R? What about uniform convergence?

Similarly trying to find the pointwise limit of

f_n(x)= -1 if -1<=x<=-1/n
nx if -1/n<=x<=1/n
1 if 1/n<=x<=1

and whether or not (f_n(x)) converges uniformly to the pointwise limit is hurting my brain ...could anyone guide me through how these are done? I'm just getting really confused about how which part of the function we use for a given x seems to change as we let n tend to infinity...?

 

[Ishuda]

Hi guys, so I have just been doing some problems on point-wise convergence of functions which has been going fine. However, suddenly the questions on my practice sheets have changed in style to defining f_n(x) piecewise with the pieces dependant on a relation between x and n:

i.e

f_n(x)= 0 if x<=n
x-n if x> n.

This is really confusing me...I have no problems using the function by itself...I.e f_2(x)= 0 if x<=2 and x-2 if x>2. But I'm getting really confused about how to workout the pointwise limit...I mean as n-> infinity given a particular x it will always eventually be less than n, so is the pointwise limit just f(x)=0 for all x in R? What about uniform convergence?

Similarly trying to find the pointwise limit of

f_n(x)= -1 if -1<=x<=-1/n
nx if -1/n<=x<=1/n
1 if 1/n<=x<=1

and whether or not (f_n(x)) converges uniformly to the pointwise limit is hurting my brain ...could anyone guide me through how these are done? I'm just getting really confused about how which part of the function we use for a given x seems to change as we let n tend to infinity...?

Let's do the first one: You are correct about the point wise convergence so look at uniform convergence. The point is the series has to converge for every x, independent of x. That is there is some N so that, for all x and n > N, |f_n(x) - f(x)| is as small as we want to make it, i.e. less than $\displaystyle \epsilon$. So assume there is such an N for $\displaystyle \epsilon$ = 1. Thus f(x) exists. Now, we know the point wise limit has to be the uniform limit, so f(x) = 0 and |f_n(x)| has to be less than 1 for n > N if the series converges uniformly. What happens if x = N+100 or N+2 or ...?

 

[Nemesis10192]

Let's do the first one: You are correct about the point wise convergence so look at uniform convergence. The point is the series has to converge for every x, independent of x. That is there is some N so that, for all x and n > N, |f_n(x) - f(x)| is as small as we want to make it, i.e. less than $\displaystyle \epsilon$. So assume there is such an N for $\displaystyle \epsilon$ = 1. Thus f(x) exists. Now, we know the point wise limit has to be the uniform limit, so f(x) = 0 and |f_n(x)| has to be less than 1 for n > N if the series converges uniformly. What happens if x = N+100 or N+2 or ...?

Well if x=N+100 then x>N so we use the x-N part so |fN(x)|=N+100 -N=100>1 so so the convergence is NOT uniform? I know the definition of uniform convergence, but am just struggling to get my head around these examples. What about the second one? If that is correct for the first example, how would you write a formal solution? I'm still confused exactly whats going on .

 

[Ishuda]

Well if x=N+100 then x>N so we use the x-N part so |fN(x)|=N+100 -N=100>1 so so the convergence is NOT uniform? I know the definition of uniform convergence, but am just struggling to get my head around these examples. What about the second one? If that is correct for the first example, how would you write a formal solution? I'm still confused exactly whats going on .

Correct, the convergence is not uniform for the first function. For the second function, let's condense it a bit to start with. We can actually restrict ourselves to the positive real values of x since the negative reals values would just be given by f(-x) = - f(x); x belongs to (-1, 0).

So, now go back to the definition for uniform convergence again. The point wise limit is 1 but if the series is uniformly convergent then there existis an N so that ... Let $\displaystyle \epsilon = \frac{1}{2}$. What happens if x is smaller than that 1/N?