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jeff1evesque

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## Homework Statement

Suppose [tex]f_n : [0, 1]\rightarrow R[/tex] is continuous and lim[tex]_{n \rightarrow \infty}f_n(x)[/tex] exists for each x in [0,1]. Denote the limit by [tex]f(x)[/tex].

Is f necessarily continuous?

## Homework Equations

We know by Arzela-Ascoli theorem:

If [tex]f_n: [a,b] \rightarrow R[/tex] is continuous, and [tex]f_n[/tex] converges to [tex]f [/tex]uniformly, then [tex]f[/tex] is continuous.

## The Attempt at a Solution

Question: Does the fact of knowing

give us insight to declare that [tex]f_n[/tex] converges to [tex]f[/tex] uniformly- and thus satisfying Arzela-Ascoli's theorem?lim[tex]_{n \rightarrow \infty}f_n(x)[/tex] exists for each [tex]x \in [0,1][/tex]. Denote the limit by f(x).

Thanks,Jeffrey Levesque

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