Determine which of the following functions are uniformly continuous on the open unit interval (0,1) Could someone help me through this problem?
Determine which of the following functions are uniformly continuous on the open unit interval $(0,1)$ : 
a) $1/(1-x)$
b) $1/(2-x)$
c) $\sin{x}$
d) $\sin{1/x}$
e) $x^3$
 A: Well, $x \mapsto \sin x$ can be extended continuously to $[0,1]$, and therefore it is uniformly continuous on $(0,1)$. The same for $x \mapsto x^3$. The function in (a) has a vertical asymptote $x=1$, and it is very easy to check that it cannot be uniformy continuous. Formally, choose $x_n=1-\frac{1}{n}$ and $y_n=1-\frac{1}{n+1}$. Then $x_n \to 1$, $y_n \to 1$, but $$\frac{1}{1-x_n}-\frac{1}{1-y_n}=n-(n+1)=-1.$$ Try to understand what happens in case (b).
More generally, it is a nice exercise to prove the following: let $f \colon [a,b) \to \mathbb{R}$ a continuous function. If $\lim_{x \to b-} f(x) = \pm \infty$, then $f$ can't be uniformly continuous. Hint: otherwise, there would exist a continuous extension $\tilde{f} \colon [a,b] \to \mathbb{R}$ of $f$, namely $\tilde{f}(x)=f(x)$ for every $x \in [a,b)$.
A: One way in which I check uniform continuity of function is the following: Given an interval $[a,b]$ look at points where the function increases rapidly, for example consider $f(x)=\frac{1}{x}$ on $(0,1)$. This function can't be uniformly continuous because as $x \to 0, \frac{1}{x} \to \infty$ and hence you can't have $|f(x)-f(y)| &lt \epsilon$ for all $x \in (0,1)$. 
Another useful way of checking uniform continuity of differentiable functions to look at:"$\text{their derivative and see if they are bounded in that given interval.}$" For example, $f(x)=x^{3}$ on $(0,1)$ has derivative $3x^{2}$ and attains a maximum of $3$ as $x \to 1$. So $f$ here is bounded by $3$. So this function has to be uniformly continuous. Similarly functions such as $\sin{x},\cos{x}$ are uniformly continuous on any given interval. But functions such as $f(x)=x^{2},x^{3}$ aren't uniformly continuous over $\mathbb{R}$ but they are $\text{uniformly continuous}$ over any interval. 
