Uniformly convergent sequence of functions in (0,1) that do no converges uniformly in [0,1]

Question

Does there exist a uniformly convergent sequence of functions in (0,1) that does not converges uniformly on [0,1]?

Answer 1

Edit : the answer below is for another question than the one really asked (it was not clear at first):

Let $f_n$ be a sequence of function that converge uniformly on $]0,1[$ and pointwise on $[0,1]$. Does it converge uniformly on $[0,1]$?.

And the answer is this question is yes :

Let $$a_n = \|f_n-f\|_{\infty, ]0,1[}$$ $$b_n = |f_n(0)-f(0)|$$ and $$c_n = |f_n(1)-f(1)|$$

Then

$$\|f_n-f\|_{\infty, [0,1]} = d_n = \max (a_n,b_n,c_n)$$

And as the three sequence converge to 0, the max of the three converge to 0

Answer 2

Yes, let $$f_n(x) = \left\{\begin{array}{cc} (-1)^n, && x=0\text{ or }1\\ 0, && x\in(0,1)\end{array}\right.$$