# Thread: any example of sequence of continuous function

1. ## any example of sequence of continuous function

fn is a sequence of uniformly continuous real-valued functions on R.

and fn converges pointwise to f. which is continuous but not uniformly continuous.

2. ## give example of a continuous function

f:[0, infty) ----> R is continuous.

f(x) tends to a finite limit as x-->infty

must f be uniformly continuous on [0,infty)?

any counterexample

3. Originally Posted by szpengchao
f:[0, infty) ----> R is continuous. f(x) tends to a finite limit as x-->infty must f be uniformly continuous on [0,infty)?
We can assume that $\lim _{x \to \infty } f(x) = L$
In this case, $\varepsilon > 0 \Rightarrow \left( {\exists N} \right)\left[ {x \geqslant N \Rightarrow \left| {f(x) - L} \right| < \frac{\varepsilon }{2}} \right]$.
Now I ask you is $f$ uniformly continuous on $\left[ {0,N} \right]$? (Why or Why Not?)
So what is your answer?

4. ## no

no, i think. for that function to be uni. continuous on [a,b], we need:

$\forall \epsilon>0, \ \exists \delta>0, \forall x,y\in[0,N], \ |x-y|<\delta\Rightarrow |f(x)-f(y)|<\epsilon$

and f(y) takes value L at $y\rightarrow\infty$
and therefore we cannot have an arbitary small distance between x,y.

i appreciate your way of "teaching."

5. If a function is continuous on a compact set, such as [0,N], is it uniformly continuous on that set?

yes