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

• Nov 12th 2010, 10:46 PM
mathematicalbagpiper
Sequence Of Bounded Functions That Converge Pointwise To An Unbounded Function
Give an example of a sequence of bounded functions $\displaystyle f_n:[0,1]\longrightarrow\mathbb{R}$ that converge pointwise to an unbounded function $\displaystyle f$.

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 $\displaystyle f(x)=\frac{1}{x}$ but that function is not defined at 0.

Am I even thinking right on this?
• Nov 13th 2010, 12:44 AM
Opalg
Quote:

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

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 $\displaystyle f(x)=\frac{1}{x}$ 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
$\displaystyle f(x) = \begin{cases}1/x&\text{if }x>0,\\ 1&\text{if }x=0.\end{cases}$

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