# few multiple choice question on continuity

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.

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