# show that the function is continuous?

• Apr 26th 2010, 08:32 AM
tn11631
show 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 PM
Amer
Quote:

Originally Posted by tn11631
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