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which of the following statements are true

  1. let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\sin x^3$. then $f$ is continuous but not uniformly continuous.

  2. every differentiable function $f:(0,1)\to[0,1]$ is uniformly continuous.

  3. $f:X\to Y$ be a continuous map between metric spaces. if $f$ is a bijection, then its inverse is also continuous.

my thoughts:

  1. true as its derivative is unbounded.

  2. false example is $x^2\sin(1/x)$.

  3. false.

please somebody confirm me about my thinkings. thank you.

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1 Answer

up vote 0 down vote accepted

Your conclusions are correct, but you haven't given correct reasons.

  1. There are uniformly continuous functions with unbounded derivatives, like $\sqrt[3]{x}$. (There are also examples where the derivative is everywhere defined.)

  2. That example is uniformly continuous.

  3. Can you think of an example?

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can u suggest me proper counter example or proofs for the problem. –  poton Dec 10 '12 at 4:45
    
for 3 i think x=1-x do the job –  poton Dec 10 '12 at 4:46
    
@poton: I don't understand your last comment. For (3) I recommend wrapping a half-open interval around a circle. For (1) I recommend showing that $\sqrt[3]{n\pi}$ and $\sqrt[3]{n\pi+\frac{\pi}{2}}$ get arbitrarily close as $n$ increases, but applying $f$ at these points gives outputs $1$ apart. For (2) I recommend choosing an example that does not have a limit at $0$. –  Jonas Meyer Dec 10 '12 at 4:49
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