# Thread: example of sequence of functions

1. ## example of sequence of functions

Can anyone think of an example of a sequence of functions $f_n:[0,1]\to\mathbb{R}$ that converges uniformly on $(0,1)$ but not on $[0,1]$?

2. I assume the functions need to be continuous, in which case no such sequence exists. The behavior of a continuous function is determined by its behavior on any dense subset.

3. Originally Posted by retep
Can anyone think of an example of a sequence of functions $f_n:[0,1]\to\mathbb{R}$ that converges uniformly on $(0,1)$ but not on $[0,1]$?

If we don't assume $f_n$ need to be continuous a trivial example is:

$f_n(x)=\begin{Bmatrix} 0 & \mbox{ if }& x\in [0,1)\\(-1)^n & \mbox{if}& x=1\end{matrix}$

Fernando Revilla

4. can someone help me , i now what does mean the sequence of real number , but i confused what does it mean by sequence of vector space? for example vector space of polynomials ? and n*m matrices ?how would be their sentences of the related sequences? and why the convergent of these sequences depends on the norm that we define for these spaces ?