I have to find if the following two functions are or not uniformly continuous:
$a) \ f:[0,1]\rightarrow R, f(x)=\frac{1}{x^2-x-2}$
$b) \ f:R\rightarrow R,f(x) = \sin(x) - cos(x)$
My way of thinking for each of them goes as follows:
a) $f$ is continuous on [0,1] and [0,1] is a compact set, thus $f$ is uniformly continuous
b) $f$ id derivable and $f'(x)=\cos(x) - \sin(x)$; however both $sin$ and $cos$ lie somewhere in [-1,1] and so does $\cos - \sin$, thus $f'$ is bounded. This makes $f$ a $Lipschitz$ function and so it is uniformly continuous.
All I'm asking is if this is correct and if not, could anyone give the proper solution? I'm asking this because the way I solved them seems to easy to be ok.