# show that the function is continuous?

• Apr 26th 2010, 09:32 AM
tn11631
show that the function is continuous?
How would i show that f(x)= $x^3$-3 $x^2$+17 is continuous on [-1,1]?
• Apr 27th 2010, 01:52 PM
Amer
Quote:

Originally Posted by tn11631
How would i show that f(x)= $x^3$-3 $x^2$+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

$g(x) = x^3$ , $h(x) = -3x^2$ , $j(x) = 17$

I will prove the first one and leave the others for you

given $\epsilon >0$ and take $c \in [-1,1]$

we need to find delta depending on epsilon and x

$\mid x^3 - c^3 \mid$

$\mid (x-c)(x^2+cx+c^2) \mid$

$\mid (x-c)(x^2+cx+c^2) \mid \leq \mid x-c \mid \mid (x^2+cx+c^2) \mid$

$\mid x-c \mid \mid (x^2+cx+c^2) \mid < \epsilon$

$\delta \mid (x^2+cx+c^2) \mid < \epsilon$

choose $\delta = \frac{\epsilon }{(x^2+cx+c^2)}$

this delta works for any epsilon