Which of the following are uniformly continuous on $(0,1)$

Question

$e^x , x, \tan(\frac{\pi x}{2}), \sin(x)$.

x and $\sin x$ are unf continuous but I dont know about other two. I guess they wont be as they go to infinity.

Answer 1

$e^x$, $\sin x$, and $x$ are uniformly continuous in $(0,1)$ because they can be continuously extended to $[0,1]$. A consequence of a theorem you should know is that if $f$ is continuous in $[0,1]$, then it is uniformly continuous in $(0,1)$.

$f(x)=\tan \frac{\pi}{2} x$ is not uniformly continuous because it blows up as $x \to 1$. Indeed, for all $\delta>0, f[(1- \delta,1)] = (\tan (\frac{\pi}{2}(1 - \delta)), +\infty)$ since $\tan$ is increasing; in this set we can find numbers arbitrarily far away from each other.