Originally Posted by

**cp05** for the second part, what about when f and g are bounded? i know when the derivative is bounded they are uniform continuous because it will be a lipschitz function, but I haven't read anything about f and g being bounded

If f and g are both uniformly continuous on R, and are both bounded, then fg will be uniformly continuous, because

$\displaystyle \begin{aligned}|f(x)g(x)-f(y)g(y)| &= |f(x)\bigl(g(x)-g(y)\bigr) + \bigl(f(x)-f(y)\bigr)g(y)| \\ &\leqslant |f(x)||g(x)-g(y)| + |f(x)-f(y)||g(y)| \end{aligned}

$

and you can make the right-hand side of that inequality as small as you want, for x and y sufficiently close together, using the given properties of f and g.