1. ## Continuous function

prove using the $\displaystyle \epsilon , \delta$ definition of continuity that $\displaystyle f(x) = x^3 - 2x^2 + 1$ is continuous at $\displaystyle x=3$

this is the start of my solution, however i need help complting it

$\displaystyle f(x) = x^3 -2x^2 +1$
$\displaystyle c=3$
$\displaystyle f(c)= 27-18+1 = 10$

for each positive number $\displaystyle \epsilon$
$\displaystyle |f(x) -f(c)| < \epsilon$ for all $\displaystyle x$ satisfying $\displaystyle |x-c|<\delta$

and then i get

$\displaystyle |x-3||x^2 + x +3|<\epsilon$ for all $\displaystyle |x-3|<\delta$

so my question is what is the value of$\displaystyle \delta$, can i just use any random number or does it have to be specific?

2. Originally Posted by iLikeMaths
prove using the $\displaystyle \epsilon , \delta$ definition of continuity that $\displaystyle f(x) = x^3 - 2x^2 + 1$ is continuous at $\displaystyle x=3$

this is the start of my solution, however i need help complting it

$\displaystyle f(x) = x^3 -2x^2 +1$
$\displaystyle c=3$
$\displaystyle f(c)= 27-18+1 = 10$

for each positive number $\displaystyle \epsilon$
$\displaystyle |f(x) -f(c)| < \epsilon$ for all $\displaystyle x$ satisfying $\displaystyle |x-c|<\delta$

and then i get

$\displaystyle |x-3||x^2 + x +3|<\epsilon$ for all $\displaystyle |x-3|<\delta$

so my question is what is the value of$\displaystyle \delta$, can i just use any random number or does it have to be specific?
Assumming your algebra is correct (I did not check it)
$\displaystyle \begin{gathered} \left| {x - 3} \right| < 1\; \Rightarrow \;2 < x < 4 \hfill \\ \left| {x^2 + x + 3} \right| \leqslant \left| {x^2 } \right| + \left| x \right| + \left| 3 \right| < 16 + 4 + 3 = 23 \hfill \\ \delta = \min \left\{ {1,\frac{\varepsilon } {{23}}} \right\} \hfill \\ \end{gathered}$