Originally Posted by
TheEmptySet The first one is not uniformly continous( I don't think)
Here is number two
Since g(x) is differentable $\displaystyle g'(x)=-4x^3e^{-x^4}$ and
the derivative is bounded on all of $\displaystyle \mathbb{R}$.
so $\displaystyle f'(x) < M $ for some $\displaystyle M \in \mathbb{R}$
let $\displaystyle \epsilon > 0$ set $\displaystyle \delta=\frac{\epsilon}{M}$
So now if $\displaystyle |x-y|< \delta$
We need to show that $\displaystyle |f(x)-f(y)|< \epsilon$
Now by the Mean Value theorem on $\displaystyle [x,y]$
$\displaystyle f(x)-f(y)=f'(c)(x-y)$
$\displaystyle |f(x)-f(y)|=|f'(c)(x-y)|=|f'(c)||x-y|=f'(c)\cdot \delta =f'(c) \frac{\epsilon}{M}< \epsilon$