How would i show that f(x)=$\displaystyle x^3$-3$\displaystyle x^2$+17 is continuous on [-1,1]?
there is a theorem said if "f" and "g" continuous functions then f+g is continuous
so I'd like to split the function into three functions
$\displaystyle g(x) = x^3 $ , $\displaystyle h(x) = -3x^2 $ , $\displaystyle j(x) = 17 $
I will prove the first one and leave the others for you
given $\displaystyle \epsilon >0 $ and take $\displaystyle c \in [-1,1] $
we need to find delta depending on epsilon and x
$\displaystyle \mid x^3 - c^3 \mid $
$\displaystyle \mid (x-c)(x^2+cx+c^2) \mid$
$\displaystyle \mid (x-c)(x^2+cx+c^2) \mid \leq \mid x-c \mid \mid (x^2+cx+c^2) \mid $
$\displaystyle \mid x-c \mid \mid (x^2+cx+c^2) \mid < \epsilon$
$\displaystyle \delta \mid (x^2+cx+c^2) \mid < \epsilon$
choose $\displaystyle \delta = \frac{\epsilon }{(x^2+cx+c^2)} $
this delta works for any epsilon