# Continuity and liimit of functions

1. Dec 12, 2009

### jeff1evesque

1. The problem statement, all variables and given/known data
Suppose $$f_n : [0, 1]\rightarrow R$$ is continuous and lim$$_{n \rightarrow \infty}f_n(x)$$ exists for each x in [0,1]. Denote the limit by $$f(x)$$.

Is f necessarily continuous?

2. Relevant equations
We know by Arzela-Ascoli theorem:
If $$f_n: [a,b] \rightarrow R$$ is continuous, and $$f_n$$ converges to $$f$$uniformly, then $$f$$ is continuous.

3. The attempt at a solution
Question: Does the fact of knowing
give us insight to declare that $$f_n$$ converges to $$f$$ uniformly- and thus satisfying Arzela-Ascoli's theorem?

Thanks,

Jeffrey Levesque

Last edited: Dec 12, 2009
2. Dec 12, 2009

### Dick

Well, no. They only gave you that the fn are continuous and converge pointwise. I think they want you find an example of a sequence of continuous functions that are pointwise convergent but don't have an continuous limit. Can you think of one?

3. Dec 12, 2009

### jeff1evesque

Can someone provide some insight for me as to what the following means:

And how I could use this fact to construct my justification for whether f is necessarily continuous?

4. Dec 12, 2009

### Dick

f isn't necessarily continuous. Face it. It says fn converges at each point. That's not enough to prove f is continuous.

5. Dec 13, 2009

### HallsofIvy

Dick is suggesting that you find a counter-example. Taking fn to be piecewise linear will suffice.

6. Dec 13, 2009

### Landau

The Arzela-Ascoli theorem asserts something about a sequence of equicontinuous functions. This has little to do with your question [or you have seen a different version of A-A].

Just construct a counter-example, i.e. a sequence of continuous functions (f_n)_n which converges pointwise to some discontinuous function. (Every book that introduces the concept of 'uniform convergence' will have such a counter-example, so you probably have encountered one already.)