Clarity on Question About Continuity and Uniform Continuity

Question

Can someone help clarify this question for me?

For each function, determine whether the function is continuous or not, or uniformly continuous or not, on each of the three intervals: $(0,1)$, $[0,1]$, $[2,\infty)$. The functions are $$f(x) = \cos(x),\quad g(x) = e^x ,\quad h(x) = \frac{1}{1-x}.$$

Would I not just simply need to find if each function is continuous/uniformly continuous on the entire real line? For example, I know $\cos(x)$ is uniformly continuous on all of $\mathbb{R}$ while $e^x$ is not uniformly continuous on all of but it is continuous. Would these statements not simply transfer over to the given intervals?

Answer 1

No. For instance, $\exp$ is not uniformly continuous on $\mathbb R$, but it is uniformly continuous on any bounded interval. Actually, it's also uniformly continuous on any interval of the form $(-\infty,a]$.

Answer 2

A function that is uniformly continuous on all of $\mathbb R$ is also uniformly continuos in any sub intervals.

A function that is continuous on all of $\mathbb R$ is also continuos in any sub intervals.

A function that is continuos but not uniformly continuos on $\mathbb R$ if restricted to a certain sub interval can become uniformly continuos, for example $e^x$ is unif. cont. in $[0,1]$