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Let $f:\mathbb R\to\mathbb R$ be continuous. Suppose $(x_n)_n$ and $(y_n)_n$ are sequences in $\mathbb R$ such that the sequence $(x_n-y_n)_n$ converges to $0$. Does this mean that the sequence $(f(x_n)-f(y_n))_n$ converges to $0$?

I feel like it is true, since the definition of continuity states that $f$ preserves limits of convergent sequences, but I do not know how to prove it.

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    $\begingroup$ Try thinking about continuous functions that are not uniformly continuous. Based on your remark, to find a counterexample you should make sure that $(x_n)$ and $(y_n)$ are not convergent sequences. $\endgroup$ – Jonas Meyer May 22 '12 at 15:46
  • $\begingroup$ @Jonas Meyer thank you for your help, I've just found a counterexample. $\endgroup$ – D. Didier May 22 '12 at 15:55
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Try $f(x) = e^{x^2}$, with $y_n = n$, $x_n = n+\frac{1}{n}$. Then $(x_n-y_n) \to 0$, but $f(x_n)-f(y_n) = e^{n^2}(e^{2+\frac{1}{n^2}}-1)$, which is unbounded.

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    $\begingroup$ $x^2$ works, too, & is a little easier. $\endgroup$ – Jonas Meyer May 22 '12 at 15:48
  • $\begingroup$ Didn't think of it! $\endgroup$ – copper.hat May 22 '12 at 15:49
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  • The result is true is $f$ is uniformly continuous on $\Bbb R$.
  • But if it's not the case, we can find $\{x_n\}$ and $\{y_n\}$ which contradict this fact, using the definition of uniform continuity.
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