Hey I am having trouble with these:



I don't really have the intuition to determine if these are uniformly convergent or not, and we didn't get much theorems characterizing uniform convergence. I am having trouble using the definition to prove they are uniformly convergent (if they even are) i.e. finding the correct choice of N such that |fn(x) - f(x)| < epsilon for n >= N. The pointwise limits that I got were (not sure if they are correct) :

i) 1
ii) does not exist on [a,b], 0 on R
iii) 0

(i) point-wise limit is 1 over ]0;1], and 0 for 0.

This is suffisant to see that the limit is not uniform.

There's a theorem claiming that : if (f_n) is continuous for every n, and if (f_n) converges uniformly to f, then f is continuous.
Here, the limit f is discontinuous at 0, proving that the limit is not uniform.

(ii) limit is 0, wether on [a;b] or on R.

On [a;b], the convergence is uniform : as soon as n > b, every f_n is null.
On R, the convergence is not uniform : take any ε > 0, it's impossible to find N such that | f_n(x) - 0 | < ε for every x in R and every n > N.

Indeed, for every n, | f_n(x) | approaches + ∞ as x tends to + ∞.

(iii) Point-wise limit is 0, indeed.
Here again, convergence is not uniform, for the same argument : lim (f_n(x)) = +∞ as x → ∞.

Thank you very much!