Determine whether the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$.
I see that the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$. Am I correct?
My answer: since the function $\cos (1/x)$ is continuous on $\mathbb{R} - \{0\}$ so it is continuous on any closed interval not containing $0$ say $[a,b]$, $a\neq 0$. Since $e^x$ is continuous for all $x$ then we can say that $f(x) = e^x \cos (1/x)$ is continuous on any interval $[a,b]$ such that $a\neq 0$. So $f$ is continuous on any subset of $[a,b]$ so it is uniformly continuous using the theorem that: A continuous function on an interval $[a,b]$ is uniformly continuous .
Is my solution correct? If not could you please provide me with the correct solution?
No, you are not correct.
Hint. Any uniformly continuous function on $(0,1)$ can be extended to an (uniformly) continuous function on $[0,1]$. See for example HERE.