Suppose I want to check if $f(x)$ is uniform continuous on a bounded interval $I$ (for eg open interval $(0,1)$), given that it is continuous on $I$. How do I do that?
My approach: Take $\bar{I}$, then two case can happen:
Case I: If I can continuously extend the function, then $f(x)$ is uniformly continuous on $I$.
Case II: If I cannot extend the function continuously, then two sub cases are possible
Subcase II a: $f(x)$ is tends to an infinite limit i.e. it shoots up/down arbitrarily for eg functions like $\frac{1}{x}$. In which case I conclude that $f$ is not uniformly continuous on $I$.
Subcase II b: $f(x)$ doesn't have a limit i.e. function of the type sin$\frac{1}{x}$. In this case as well $f(x)$ is not uniform continuous on $I$.
So is my above classification of continuous function sufficient to determine which functions are uniform continuous and which are not? So far it had worked well for me.
Best Answer
Yes, that is correct. In fact, assuming that the domain of $f$ is $(a,b)$: