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Let $x_{n}$ be a sequence of continuous functions uniformly convergent to the function $x$, the domain of all functions $x_{n}$ be an interval $[a,b]$ and let $g_{n}(t):=f(t,x_{n}(t))$ and $g(t):=f(t,x(t))$, where $f$ is real-valued function with open domain.

Question: What assumptions about $f$ should be made to get the uniform convergence $g_{n}(t)$ to $g(t)$ on the interval $[a,b]$?

I suspect that continuity of $f$ is not sufficient, probably one can assume the uniform continuity of $f$, but I can not prove any of these claims.
I would be very grateful for any hints.

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2 Answers

I think that the necessary uniformly continuity of $f$ in such a case is obtained automatically on our set of interest(it is my point 2 following from Cantor's theorem as stated in Lang's Real and Functional Analysis).

Below I sketch how continuity of $f$ is sufficient:

  1. $C=\{(t,x(t)):t\in[a,b]\}$ is compact (for the continuity of $x$), and it is contained in the domain of definition of $f$ (under your assumption);
  2. $f$ is continuos so for Cantor's theorem it is uniformly continuous relatively at $C,$ this means that for any $\varepsilon$ there exists $\delta$ such that $$\|(t',x')-(t'',x'')\|<\delta\textrm{ and }(t',x')\in C\Rightarrow |f(t'',x'')-f(x',t')|<\varepsilon.$$
  3. $\{x_n\}_{n\in\mathbb{N}}$ is uniformly convergent towards $x,$ that means for any $r$ there exists $\nu\in\mathbb{N}$ s.t. $\|x_n-x\|<r$ for any $n>\nu.$

From 1.,2., and 3. we conclude easily the uniform convergence of $\{f\circ(\textrm{id},x_n)\}_{n\in\mathbb{N}}$ towards $f\circ(\textrm{id},x)$ on $[a,b].$

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Thank you Giuseppe for your answer. –  John Apr 29 '12 at 10:05
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I think you may need a Hölder or Lipschitz uniform continuity condition of some order $\alpha > 0$ to get this. But when you do it comes out so easily that you may wonder if uniform continuity of $f$ -on this open domain you mentioned - wouldn't be enough.. I'll have to look for counterexamples. I don't think there are necessary conditions on $f$, because you might 'tweak' an example of $x_n$'s given a particular $f$ where say $g_n(t)$ was identically constant or something like that. This is why I say your question 'what conditions should be made about $f'$ is somewhat misleading, really it's 'what statements I can prove about f will guarantee this result'. If I can find some counterexamples where uniform continuity is not enough, etc. I will work on it.(later that AM) I see that if $x_n\to x$ uniformly and if $f$ is merely continuous, then we have pointwise convergence of $g_n\to g$ (all continuous too). But this can allow for some absurdities(maybe not if your paths were in $\mathbb R^2$ or less to begin with).

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Thank you James. May I ask if you found an counterexample where uniform continuity is not enough? For me it's hard to imagine that such a counterexample exists... –  John Apr 29 '12 at 10:07
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