How would i show that f(x)=$\displaystyle x^3$-3$\displaystyle x^2$+17 is continuous on [-1,1]?

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- Apr 26th 2010, 08:32 AMtn11631show that the function is continuous?
How would i show that f(x)=$\displaystyle x^3$-3$\displaystyle x^2$+17 is continuous on [-1,1]?

- Apr 27th 2010, 12:52 PMAmer
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