Uniformly Continuous If $f$ is uniformly continuous on $(0,1)$, can f be extended to $[0,1]$ in such a way as to be right continuous at $0$ and left continuous at $1$ ? Justify your answer.
I was trying to give a counterexample such that $f(x) = x^3$ and take the limit from the left and right.
would someone help me out with this question .
 A: Let me help you in interpreting what the question is asking.  I think once you can understand fully what the question is asking, you will be able to come up with the solution on your own.
We know $f$ is a function with domain $(0,1)$, and $f$ is uniformly continuous on this domain.  So, if you were to draw the graph of the function on the $XY$-plane, you would only draw it over the interval $(0,1)$ on the $x$-axis.  It's also continuous, so you can never "pick up your pencil" when you are drawing it.
Now, this function takes values from $(0,1)$ to $\Bbb R$.  Every element of $(0,1)$ is sent to some real number in $\Bbb R$.
The question now is: what happens to the function as you approach $0$ from the right?  Does it approach a real number?  If not, then it must either approach $\infty$ or $-\infty$.  If it approaches a real number, then we can extend $f : (0,1) \to \Bbb R$ to a new function $\hat{f} : [0,1) \to \Bbb R$ by defining $\hat{f}(x) = f(x)$ if $x \in (0,1)$, and $\hat{f}(0) = $ the limit of $f(x)$ when we approach $0$.
Similarly, if the limit of the function as we approach $1$ from the left is also a real number, we could extend $f$ to $\hat{f} : [0,1] \to \Bbb R$, by just defining $\hat{f}$ as $f$ on $(0,1)$ and the limits on $x = 0$ and $x = 1$.
But this only works if the limits are real numbers.  But we could also have $\lim \limits_{x \to 0^{+}} f(x) = \infty$ (or $-\infty$).  And similarly, we could have $\lim \limits_{x \to 1^{-}} f(x) = \infty$ (or $-\infty$), in which case we would never be able to extend $f$ to $[0,1]$ and keep it being continuous.
So the question for you is: in the case where one of the limits mentioned in the last paragraph is $\infty$ (or $-\infty$), can $f$ still be uniformly continuous?  It could be continuous, but can it be uniformly continuous?  If not, then that means uniform continuity on $(0,1)$ would imply the limits are finite as we approach $0$ and $1$, so that we could extend $f$.
So at the end of the day, you have to determine whether if we have a function $f(x)$ which satisfies $\lim \limits_{x\to 0^{+}} f(x) = \infty$, can this function also be uniformly continuous?
