1. ## Continuous function

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

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

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

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

and then i get

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

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

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

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

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

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

and then i get

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

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