If f+g is uniformly continuous on a subset A of R then are f and g uniformly continuous??

Question

I have been going throught Bartle Sherbert book and I found the problem where it was asked to prove that f+g is uniformly continuous and so on. I was thinking about the converse of that statement. Is it true or there is some counterexample?Same with the fg continuity where it is given that f and g are bounded Thanks in advance.

Answer 1

Take $g=-f$ Do you believe that not every function is uniformly continuous? For the second question take a positive function $f$ which is not uniformly continuous and take $g=\frac 1 f$.

Answer 2

Look for example at the Dirichlet function $\varphi: \mathbb{R} \longrightarrow \mathbb{R}$ with $\varphi(x)=0$, if $x \in \mathbb{Q}$ otherwise $\varphi(x)=1$.

Then you can look at $\varphi - \varphi=0$ and this is uniformely continuous but $\varphi$ is not even continuous.