# Thread: Sequence Of Bounded Functions That Converge Pointwise To An Unbounded Function

1. ## Sequence Of Bounded Functions That Converge Pointwise To An Unbounded Function

Give an example of a sequence of bounded functions $f_n:[0,1]\longrightarrow\mathbb{R}$ that converge pointwise to an unbounded function $f$.

OK, so here's what I'm thinking. I want something that has nice behavior everywhere except for one point, where I want things to kind of go sour. I was thinking along the lines of converging to $f(x)=\frac{1}{x}$ but that function is not defined at 0.

Am I even thinking right on this?

2. Originally Posted by mathematicalbagpiper
Give an example of a sequence of bounded functions $f_n:[0,1]\longrightarrow\mathbb{R}$ that converge pointwise to an unbounded function $f$.

OK, so here's what I'm thinking. I want something that has nice behavior everywhere except for one point, where I want things to kind of go sour. I was thinking along the lines of converging to $f(x)=\frac{1}{x}$ but that function is not defined at 0.

Am I even thinking right on this?
Yes, that is along the right lines. As you say, the only trouble is that the limit function needs to be defined at 0. So you might look for a sequence of functions that converges pointwise to a function such as
$f(x) = \begin{cases}1/x&\text{if }x>0,\\ 1&\text{if }x=0.\end{cases}$

Spoiler:
What about $f_n(x) = \dfrac1{x + (1-x)^n}$ ?